But with these crude materials there went an abundance of time, so that a number of great results were accomplished in spite of the difficulties attending the study of the subject. It is said that Hippocrates of Chios (_ca._ 440 B.C.) wrote the first elementary textbook on mathematics and invented the method of geometric reduction, the replacing of a proposition to be proved by another which, when proved, allows the first one to be demonstrated. A little later Eudoxus of Cnidus (_ca._ 375 B.C.), a pupil of Plato's, used the _reductio ad absurdum_, and Plato is said to have invented the method of proof by analysis, an elaboration of the plan used by Hippocrates. Thus these early philosophers taught their pupils not facts alone, but methods of proof, giving them power as well as knowledge. Furthermore, they taught them how to discuss their problems, investigating the conditions under which they are capable of solution. This feature of the work they called the _diorismus_, and it seems to have started with Leon, a follower of Plato. Between the time of Plato (_ca._ 400 B.C.) and Euclid (_ca._ 300 B.C.) several attempts were made to arrange the accumulated material of elementary geometry in a textbook. Plato had laid the foundations for the science, in the form of axioms, postulates, and definitions, and he had limited the instruments to the straightedge and the compasses. Aristotle (_ca._ 350 B.C.) had paid special attention to the history of the subject, thus finding out what had already been accomplished, and had also made much of the applications of geometry. The world was therefore ready for a good teacher who should gather the material and arrange it scientifically. After several attempts to find the man for such a task, he was discovered in Euclid, and to his work the next chapter is devoted. After Euclid, Archimedes (_ca._ 250 B.C.) made his great contributions. He was not a teacher like his illustrious predecessor, but he was a great discoverer. He has left us, however, a statement of his methods of investigation which is helpful to those who teach. These methods were largely experimental, even extending to the weighing of geometric forms to discover certain relations, the proof being given later. Here was born, perhaps, what has been called the laboratory method of the present. Of the other Greek teachers we have but little information as to methods of imparting instruction. It is not until the Middle Ages that there is much known in this line. Whatever of geometry was taught seems to have been imparted by word of mouth in the way of expounding Euclid, and this was done in the ancient fashion.