was the beginning of philosophy in this sense, that it completely transformed the traditional view of the world, and made the raising of more ultimate problems inevitable. This transformation was effected in complete independence of religion. What we may call secularism was, in fact, characteristic of all eastern Ionian science to the end. We must not be misled by the fact that Anaximander called his innumerable worlds 'gods' and that his successor Anaximenes spoke of Air as a 'god'. These were never the gods of any city and were never worshipped by any one, and they did not therefore answer at all to what the ordinary Greek meant by a god. The use of the term by the Milesians means rather that the place once occupied by the gods of religion was now being taken by the great fundamental phenomena of nature, and the later Greeks were quite right, from their own point of view, in calling that atheism. Aristophanes characterizes this way of speaking very accurately indeed in the _Clouds_ when he makes Strepsiades sum up the teaching he has received in the words 'Vortex has driven out Zeus and reigns in his stead', and when he makes Socrates swear by 'Chaos, Respiration and Air'. So too the Milesians spoke of the primary substance as 'ageless and deathless', which is a Homeric phrase used to mark the difference between gods and men, but this only means that the emotion formerly attached to the divine was now being transferred to the natural. The Milesians, then, had formed the conception of an eternal matter out of which all things are produced and into which all things return, and the conception of Matter belongs to philosophy rather than to science. But besides this they had laid the foundations of geometry, and that led in other hands to the formulation of the correlative conception of Limit or Form. It is needless to enumerate here the Milesian and Pythagorean contributions to plane geometry; it will be sufficient to remind the reader that they covered most of the ground of _Euclid_, Books I, II, IV, and VI, and probably also of Book III. In addition, Pythagoras founded Arithmetic, that is, the scientific theory of numbers (αριθμητικη {arithmêtikê}), as opposed to the practical art of calculation (λογιστικη {logistikê}). We also know that he discovered the sphericity of the earth, and the numerical ratios of the intervals between the concordant notes of the octave. It is obvious that he was a scientific genius of the first order, and it is also clear that his methods included those of observation and experiment. The discovery of the earth's spherical shape was due to observation of eclipses, and that of the intervals of the octave can only have been based on experiments with a stretched string, though the actual experiments attributed by tradition to Pythagoras are absurd. It was no doubt this last discovery that led him to formulate his doctrine in the striking saying 'Things