Zeno's first proof of the impossibility of motion referred to the impossibility of passing through a fixed space. He showed that by dividing a line into an infinite number of parts an infinite number of points would be obtained and these permitted no beginning of motion. His second proof tried to show the impossibility of passing through space having movable boundaries. The story of Achilles and the tortoise illustrates this. A pursuer in a race at every interval must reach a point from which the pursued starts simultaneously. But the latter is always in advance. The third, or "resting arrow," argument showed that a moving arrow is at every instant in some one point of its track. Its movement at such instant is then equal to zero. Its track is a group of zeros. No magnitude could be framed from these. Zeno also anticipated much later philosophical discussions, like Einstein's, relating to the relativity of motion. He took for an example a moving wagon. Its movement would appear different to observers on other moving bodies going in various directions. They would see changes in rates of speed as well as in direction. Protagoras, at a subsequent date, developed this idea of relativity and showed that things are as they appear to each individual at the moment they are perceived. He summarized his teaching in the aphorism: Man is the measure of all things. The Skeptics, 200 years later, developed the Protagorean theory of relativity, and by a series of arguments attempted to prove that perceptions change not only with the different species of animate beings, but with many conditions and circumstances. It was also shown that not only man's perceptions are subject to changes, but also his opinions following from his perceptions. Another school taught that to every opinion the opposite can be opposed with equally good reasons. CHAPTER VI GOLDEN AGE OF GREEK SCIENCES Science had made a great advance as a result of the researches and theories of the atomists. A consistent mechanical theory of matter and the universe had been set forth. Science and philosophy were stripped of many of the old superstitions that had clung to them. The leading theories invented were based on logical principles. While these changes were being worked out, numerous inventions of scientific instruments and apparatus were made and systematic methods of studying science were organized. These furnished the means for still greater progress.