Alex's Adventures in Numberland - Alex Bellos [163]
Mathematicians had more success with restating the postulate in different terms. For example, the Englishman John Wallis in the seventeenth century realized that all The Elements could be proved by keeping the first four postulates as they were but by replacing the fifth postulate with the following alternative: given any triangle, the triangle can be blown up or shrunk to any size so that the lengths of the sides stay in the same proportion to each other and the angles between the sides remain unaltered. While it was quite an insight to realize that the fifth postulate could be rephrased as a statement about triangles rather than a statement about lines, it did not resolve mathematicians’ concerns: Wallis’s alternative postulate was perhaps more intuitive than the fifth postulate, though perhaps only marginally so, but it still wasn’t as simple or self-evident as the first four. Other equivalents for the fifth postulate were also discovered; Euclid’s theorems still held true if the fifth postulate was substituted by the statement that the sum of angles in a triangle is 180 degrees, that Pythagoras’s Theorem is true, or that for all circles the ratio of the circumference to diameter is pi. Extraordinary as it might sound, each of these statements is mathematically interchangeable. The equivalent that most conveniently expressed the essence of the fifth postulate, however, concerned the behaviour of parallel lines. From the eighteenth century mathematicians studying Euclid began to prefer using this version, which is known as the parallel postulate:
Ga line and a point not on that line, then there is at most one line that goes through the point and is parallel to the original line.
It can be shown that the parallel postulate refers to the geometry of two distinct types of surface, hinging on the phrase ‘at most one line’ – which is mathspeak for ‘either one line or no lines’. In the first case, illustrated by the diagram, for any line L and point P, there is only one line parallel to L (which is marked L’) that goes through P. This version of the parallel postulate applies to the most obvious type of surface, a flat surface, such as a sheet of paper on your desk.
The parallel postulate.
Now let’s consider the second version of the postulate, in which for any line L and point P not on that line there are no lines through P and parallel to L. At first it is hard to think on what type of surface this might be the case. Where on Earth…? On Earth is exactly where! Imagine, for example, that our line L is the equator, and imagine that point P is the North Pole. The only straight lines through the North Pole are the lines of longitude, such as the Greenwich Meridian, and all lines of longitude cross the equator. So, there are no straight lines through the North Pole that are parallel to the equator.
The parallel postulate provides us with a geometry for two types of surface: flat surfaces and spherical surfaces. The Elements was concerned with flat surfaces, so for 2000 years this was the main focus of mathematical enquiry. Spherical surfaces like the Earth were of less interest to theoreticians than they were to navigators and astronomers. It was only at the beginning of the nineteenth century that mathematicians found a wider theory that encompassed flat and spherical surfaces – and this happened only after they encountered a third kind of surface, the hyperbolic one.
One of the most determined aspirants in the quest to prove the parallel postulate from the first four postulates, and therefore show that it is not a postulate at all but a theorem, was Janós Bolyai, an engineering undergraduate from Transylvania. His mathematician father Farkas knew the scale of the challenge from his own failed attempts and implored his son to stop: ‘For God’s sake, I beseech you,