The first, second, and fourth books of Euclid are largely of Pythagorean origin. For solid geometry as a science we are also indebted to this sect of number-worshipers. One of them (Archytas, 428-347 B.C., a friend of Plato) was the first to apply geometry to mechanics. We see again here, as in the case of Thales, that the love of abstract thought, the pursuit of science as science, did not interfere with ultimate practical applications. Plato (429-347 B.C.), like many other Greek philosophers, traveled extensively, visiting Asia Minor, Egypt, and Lower Italy, where Pythagorean influence was particularly strong. His chief interest lay in speculation. For him there were two worlds, the world of sense and the world of ideas. The senses deceive us; therefore, the philosopher should turn his back upon the world of sensible impressions, and develop the reason. In his _Dialogues_ he outlined a course of training and study, the professed object of which was to educate a class of philosophers. (Strange to say, Plato's curriculum, planned originally for the intellectual _élite_, still dictates in our schools the education of millions of boys and girls whose careers do not call for a training merely of the reason.) Over the porch of his school, the Academy at Athens, were inscribed the words, "Let no one who is unacquainted with geometry enter here." It was not because it was useful in everyday life that Plato laid such insistence on this study, but because it increased the students' powers of abstraction and trained the mind to correct and vigorous thinking. From his point of view the chief good of geometry is lost unless we can through it withdraw the mind from the particular and the material. He delighted in clearness of conception. His main scientific interest was in astronomy and mathematics. We owe to him the definition of a line as "length without breadth," and the formulation of the axiom, "Equals subtracted from equals leave equals."