In the second edition of this book, I made a lengthy addition to note 9 to chapter 6 (pp. 248 to 253). The historical hypothesis propounded in this note was later amplified in my paper ‘The Nature of Philosophical Problems and Their Roots in Science’ (British Journal for the Philosophy of Science, 3, 1952, pp. 124 ff.; now also in my Conjectures and Refutations). It may be restated as follows: (1) the discovery of the irrationality of the square root of two which led to the breakdown of the Pythagorean programme of reducing geometry and cosmology (and presumably all knowledge) to arithmetic, produced a crisis in Greek mathematics; (2) Euclid’s Elements are not a textbook of geometry, but rather the final attempt of the Platonic School to resolve this crisis by reconstructing the whole of mathematics and cosmology on a geometrical basis, in order to deal with the problem of irrationality systematically rather than ad hoc, thus inverting the Pythagorean programme of arithmetization; (3) it was Plato who first conceived the programme later carried out by Euclid: it was Plato who first recognized the need for a reconstruction; who chose geometry as the new basis, and the geometrical method of proportion as the new method;

who drew up the programme for a geometrization of mathematics, including arithmetic, astronomy, and cosmology; and who became the founder of the geometrical picture of the world, and thereby also the founder of modern science-of the science of Copernicus, Galileo, Kepler, and Newton.