The writings of Hypsicles, Geminus, Demascius of Demascus Ptolemy and his interest in geometry Some phases of treatment of subject matter in the practical writings mentioned that have an educa- The geometric nomenclature and symbolism of Heron. His treatment of the mensuration of Brahmagupta. His differentiation of "gross” answers from exact answers. Aryabhatta's brevity of proof. Translators and preservers of Euclid and of Greek Land surveying: The Codex Arcerianus Influence of the Alexandrian school Little influence of Euclid. Quintilian's estimate of logical geometry THE TEACHING OF GEOMETRY FROM THE RISE OF THE CHRISTIAN SCHOOLS TO THE YEAR 1525 THE CHRISTIAN SCHOOLS OF THE MIDDLE AGES The works of Capella, Boethius, Isidore of Seville, Cassi- odorus, and their use in the schools The teaching of geometry in the medieval schools before The influence of Gerbert. His geometry BOOKS ON PRACTICAL GEOMETRY BEFORE THE RISE OF THE UNIVERSITIES. Books THAT INFLUENCED THE UNIVERSITY Leonardo of Pisa. Savasorda. jordanus Nemorarius. Regiomontanus. Educational significance of their THE UNIVERSITIES OF THE MIDDLE AGES Geometry taught before the introduction of Euclid' At Paris, Oxford, Prague, Vienna, Heidelberg, Cologne, THE TEACHING OF GEOMETRY FROM THE YEAR 1525 TO SOME EARLY PRINTED BOOKS ON GEOMETRY influence of Albrecht Dürer. Geometry of a single opening of the compasses Tendencies in the above works that have a bearing on GERMANY 60-73 Place of geometry in the curriculum. Interest in Euclid Geometry in the secondary schools of the sixteenth century 61-63 Geometry in the secondary schools of the seventeenth century 63–65 The influence of Comenius. The geometries of Geometry in the secondary schools of the eighteenth century 65–71 Importance of mathématics and science. The texts of Wolf, Kästner, and Sturm, and their influence. The Geometry in the secondary schools of the nineteenth century 71–72 Reorganization of the Gymnasia of Prussia. Admis- sion to universities. General nature of the teaching of geometry and mathematics in general General resume from the sixteenth century, showing the development of the teaching of geometry from the standpoint 72–73 General survey from the sixteenth century to the middle of the Early books that influenced teaching 74-78 Bouvelles. Ramus. Errard. Mercator. Arnauld. Le Clerc. The interest in the teaching of geometry up to the time of the expulsion of the Jesuits General survey. Use of texts in class work. Books of Rivard, Clairaut, and La Caille. Features with texts used in them 81-82 83-86 Suggestions on teaching by Lacroix. By Busset 86-87 In the universities 87-88 English translation of Euclid. Professorships of ' Elements edited by Barrow, Whiston, Simson, The traditions Euclidean 88–89 89-90 Before the creation of the Gymnasia The beginning practical. Influence of Peter the Great in establishing special schools where mathematics was taught. Methods employed in these Aim to develop pure science. Introduction of text- Attempts to break away from the dogmatic method. 90 Organization of public schools. Schools for training the nineteenth century The beginning practical. Books for academic teach- ing. Beginning of university requirements in mathematics. Geometry in the Gymnasia Influence of the Jesuits. The teaching of geometry in The geometry taught at Harvard,' Yale, and other universities up to the time of entrance requirements 98-99 First course of the Boston English High School in 1821. The accrediting system Influence of the English and the French Summary of the teaching of geometry since the beginning of the 100-103 PRESENT-DAY TEACHING OF GEOMETRY The place of geometry in the curriculum 104-105 105-106 The place of geometry in the curriculum 106-108 108-109 Recent reforms in the teaching of geometry 109-112 The place of geometry in the curriculum 112-115 115-116 The teaching of geometry in the elementary schools 116-117 The teaching of geometry in the two types of normal The teaching of geometry in the secondary schools 118-119 The place of geometry in the curriculum. Relation between the secondary schools and the university as 120-121 The teaching of geometry in the higher elementary schools 121-122 The teaching of geometry in the preparatory schools 122 The teaching of geometry in the secondary schools 122-125 Tendencies. The “Perry Movement” 126-128 Sweden Norway. Denmark. Austria. Bulgaria. The place of geometry in the curriculum The teaching of geometry in the high school 131-132 PRESENT PROBLEMS AND THEIR HISTORIC CONNECTIONS The practical preceded the logical. The transition into the logical was not abrupt. Tendencies to hold to the special have been marked. Geometry has been taught more and more to younger students The sequence in the subject-matter of geometry. sequence in the teaching of the mathematical subjects. Correlation of geometry with science. The question of THE EDUCATIONAL SITUATION TO-DAY PRESENT PROBLEMS IN THE TEACHING OF ELEMENTARY GEOMETRY 138–155 The balance between the logical and the practical. The sequence in the subject-matter of geometry. Summary The sequence in the different mathematical subjects. 143-146 The enrichment of the work in elementary mathematics by the introduction of phases of the higher work The enrichment of the subject-matter of geometry, by the introduction of modern geometry. The question of method. Summary of points 151-154 UNIVERS, A HISTORY OF THE TEACHING OF ELEMENTARY GEOMETRY CHAPTER I THE TEACHING OF GEOMETRY BEFORE EUCLID THE BEGINNING OF GEOMETRY AMONG PRIMITIVE PEOPLE room. In this attempt to trace the historic development of the teaching of elementary geometry, the word teaching will be used both in its widest sense and as restricted to methods of the school In the more general use of the word, we shall be concerned with the manner in which man began to formulate the science, with the additions to the subject-matter in the various epochs, and with the books written to spread this knowledge. In studying the development of the teaching of geometry in any epoch, four factors will be considered: 1. The contributions to the subject matter. 2. The text-books and books read by the learned. 3. The methods of teaching geometry. 4. The place of the subject in the curriculumn. The history of geometry as such will be considered only so far as is necessary for a foundation for the present study. Naturally this will form a large part of our information about the early teaching of the subject. Keeping in mind these four aspects of the matter, our subject will be treated first chronologically, bringing it down to the present time. Certain nodern problems will then be considered in the light of the foregoing historic material. Geometry as studied in the schools to-day has two values: (a) It is a study which has practical applications in mensuration and in the related fields of science. (6) It is a means of logical discipline. With respect to the general historic development of the subject-matter of geometry, we shall find: (a) That the practical side alone was recognized in pre-grecian geometry. 1 Where contradictions do not arise the term geometry will generally be employed for elementary geometry. I |