ABSTRACT
Euclid’s Elements codified what was known about geometry into a handful of axioms and then showed that all of geometry could be deduced from them. His achievement was impressive, but even proving simple results, like Pythagoras’ theorem, can take dozens of intermediate results. The Elements can be compared to a low-level programming language in which everything has to be spelt out. It was not until the nineteenth century that a more high-level approach to three-dimensional geometry was developed. On the basis of the work carried out by Hamilton on quaternions, about which we say more in Section 9.7, the theory of vectors was developed by Josiah Willard Gibbs (1839-1903) and Oliver Heaviside (1850-1925). Their theory is introduced in this chapter. Pythagoras’ theorem can now be proved in a couple of lines. But it should be remembered that the geometry described by Euclid and the geometry described in this chapter are one and the same: only the form of the descriptions differ. We have not attempted to develop the subject in this chapter completely rigorously, so we often make appeals to geometric intuition in setting up the algebraic theory of vectors. If you have a firm grasp of this intuitive theory then a rigorous development can be found in the first four chapters of [103], for example. On the other hand, the theory of vectors developed in this chapter itself provides an alternative axiomatization of geometry.
