PREFACE While much has been written on the history of mathematics, comparatively little attention has been given to the history of its teaching. Günther's “Geschichte des mathematischen Unterrichts im Deutschen Mittelalter bis zum Jahre 1525” and Suter's “Die Mathematik auf den Universitäten des Mittelalters" are the only prominent works that emphasize the teaching side. Among the works that treat the general history of mathematics, those of Cantor, Hankel, Gow, and Allman have been of great assistance. Much information has also been gained from miscellaneous articles, standard works on the history of education, and early texts, the latter constituting, for the most part, the original sources for this study. The material that has been utilized in the first three chapters is to be found chiefly in the standard histories of mathematics, but wherever possible the original sources have been consulted. Originality is claimed only for the selection and arrangement of this portion of the subject-matter and for the conclusions drawn. The material for the next three chapters has been gleaned very largely from the original sources. The author is under great obligation to Professor David Eugene Smith of Teachers College, Columbia University, under whose direction this dissertation has been prepared, for helpful counsel and criticism, and for rendering much of this work possible through his valuable collection of early printed books. ALVA WALKER STAMPER. New York, June, 1906. CONTENTS . . PAGE THE TEACHING OF GEOMETRY BEFORE EUCLID THE BEGINNING OF GEOMETRY AMONG PRIMITIVE PEOPLE . Intuitive stage 2–3 Stage in which principles are recognized. Ability to classify. Formation of rules. Practical use Application with logic as a basis Influences that helped to develop their geometry 4-7 Utilitarian influence. The overflow of the Nile Records.—The manuscript of Ahmes; inscriptions on the The development of the subject matter of elementary geometry 10-16 The first propositions given to geometry. The practical not neglected. First problems of construction. Development of a deductive Development of the geometry of areas. Further contributions. Nothing known of methods of proof. Geometry becomes an abstract science 13–16 Interest in geometry. The Sophists. The schools of Plato and Aristotle. The Three Problems of Antiquity and their relation to the development of the subject-matter of geometry Books written on geometry. Euclid not the first. Nature of the contributions. The “Eudemian Summary.” The founding of solid geometry by Lack of harmony between the historic sequence and that given later by Euclid Educational features of the Greek geometry In the Ionian and Pythagorean schools 16-17 17-25 The Socratic method. The old Greek’education. Plato's disposition of geometry in the curriculum 17-18 Four-fold division of the mathematical sciences . 18-19 Neglect of the study of solid geometry Tendency to hold to the special Conciseness of proofs 22–23 23-25 The notion of locus. The method of exhaus- tian. Reductio ad absurdum. Reduction. Analysis. The diorismus, or discussion Summary of the general characteristics of the Greek geometry 25–26 THE WORK OF EUCLID AND HIS INFLUENCE ON THE SUBSEQUENT TEACHING OF GEOMETRY The contributions by the earlier Greeks 27-28 Euclid's sequence of subject-matter. Points of de- parture from the Euclidean system Euclid's sequence not the historic 28-29 Hypothetical constructions debarred Euclid established a systematic method of proving Some adverse criticisms of Euclid Summary of the three chief characteristics of the “Elements” 30-31 With the Arabs. Translations 31-32 Some translations of the - Elements” THE TEACHING OF GEOMETRY FROM EUCLID TO THE His writings. Contributions to solid geometry. Value of n. Use of methods already laid down. Founda- chanical proofs. A recognition of the unity of the His study of the conic sections. Relation to later teaching of elementary geometry At Alexandria after Apollonius Applications of geometry to astronomy and surveying: Those who carried on this work. The beginnings of Extension of spherical geometry by Menelaus. Also 36-37 Interest in Euclid . 37-38 The commentators, Heron, Pappus, Theon, Proclus |