A history of the teaching of elementary geometry, with reference to present-day problems..

Preface

While much has been written on the history of mathematics, comparatively little attention has been given to the history of its teaching. Giinther's " Geschichte des mathematischen Unterrichts im Deutschen Mittelalter bis zum Jahre 1525" and Suter's "Die Mathematik auf den Universitaten 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.

PAL'

Chapter I The Teaching Of Geometry Before Euclid

Thb Beginning op Geometry Among Primitive People . . 1-4

Intuitive stage 2-3

Stage in which principles are recognized. Ability to

classify. Formation of rules. Practical use .... 3-4

Logical stage 4

Application with logic as a basis 4

The Egyptians 4-10

Influences that helped to develop their geometry .... 4-7 The pyramids. Rope Stretchers. Astronomy ^Esthetic influence. Mural decorations Utilitarian influence. The overflow of the Nile Records. — The manuscript of Ahmes; inscriptions on the

Temple at Edfu 7-9

The Greeks bepore Euclid 10-26

The development of the subject-matter of elementary geometry . 10-16

Thales and his school 10-11

The first propositions given to geometry. The practical not neglected. First problems of construction. Development of a deductive geometry of lines

Pythagoras and his school 11-13

Development of the geometry of areas. Further

contributions. Nothing known of methods of

proof. Geometry becomes an abstract science

The development of geometry at Athens .... 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 .... 13

Books written on geometry. Euclid not the first.

Nature of the contributions. The "Eudemian

Summary." The founding of soUd geometry by

Eudoxus 13-16

Lack of harmony between the historic sequence

and that given later by Euclid 16

Educational features of the Greek geometry 16-26

In the Ionian and Pythagorean schools .... 16-17

At Athens 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 Restriction as to the use of instruments . 19-20

V vi Contents

Neglect of the study of solid geometry . 20

Lettering of figures 21

Drawing of figures 21

Tendency to hold to the special 21-22

Conciseness of proofs . . 22-23

Methods of attack 23-25

The notion of locus. The method of exhaustion. Reductio ad absurdum. Reduction.

Analysis. The diorismus, or discussion

Summary of the general characteristics of the Greek geometry 25-26

Chapter II

THE WORK OF EUCLID AND HIS INFLUENCE ON THE SUBSEQUENT TEACHING OF GEOMETRY

The Elements of Eucud 27-31

The contributions by the earlier Greeks .... 27-28 Euclid's sequence of subject-matter. Points of departure from the Euclidean system 28

Euclid's sequence not the historic 28-29

The practical eliminated 29

Hypothetical constructions debarred 29

EucHd established a systematic method of proving

propositions 29-30

Some adverse criticisms of Euclid 30

Summary of the three chief characteristics of the ^'Elements" 30-31

The Later InfivUEnce of Euclid 31-33

At Alexandria 31

With the Arabs. Translations 31-32

In Europe 32-33

Some translations of the "Elements"

The use of Euclid in the medieval universities

Later editions of Euclid

Chapter III

THE TEACHING OF GEOMETRY FROM EUCLID TO THE RISE OF THE CHRISTIAN SCHOOLS

At Alexandria 34-39

Archim,edes 34-36

His writings. Contributions to solid geometry. Value of 77. Use of methods already laid down. Foundations of the integral calculus. Geometric and mechanical proofs. A recognition of the unity of the mathematical subjects

Apollonius of Perga 36

His study of the conic sections. Relation to later teaching of elementary geometry

At Alexandria after Apollonius 36-39

Applications of geometry to astronomy and surveying. Those who carried on this work. The beginnings of

trigonometry 36

Extension of spherical geometry by Menelaus. Also

earlier by Theodosius • . 36-37

Interest in Euclid . 37-38

The commentators, Heron, Pappus, Theon, Proclus

Contents vii

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 educational significance 38

The geometric nomenclature and symbolism of Heron. His treatment of the mensuration of heights and distances

The Orientals 39-40

The Hindus 39

Brahmagupta. His differentiation of "gross" answers from exact answers. Aryabhatta's brevity of proof. The use of verse

The Arabs 39-40

Translators and preservers of Euclid and of Greek learning in general

The Romans 40-42

Land surveying. The Codex Arcerianus Influence of the Alexandrian school Little influence of Euclid. Quintilian's estimate of logical geometry Summary showing the later influence on teaching ... 42

Chapter IV

THE TEACHING OF GEOMETRY FROM THE RISE OF THE CHRISTIAN SCHOOLS TO THE YEAR 1525

The Christian Schools of the Middi^e Ages 43-47

The works of Capella, Boethius, Isidore of Seville, Cassi-

odorus, and their use in the schools 43-44

The teaching of geometry in the medieval schools before

Gerbert 45

The influence of Gerbert. His geometry 45-47

Books on Practicai. Geometry before the Rise of the Universities. Books that Influenced the University

Teaching 47-50

Leonardo of Pisa. Savasorda. Jordanus Nemorarius. Robertus Anglicus. Sacrobosco. Bradwardine. Regiomontanus. Educational significance of their writings

The Universities of the Middle Ages 51-53

Geometry taught before the introduction of Euclid ... 51

The teaching of Euclid 51-53

At Paris, Oxford, Prague, Vienna, Heidelberg, Cologne, Leipzig, Bologna, and Pisa. Character of the teaching

Chapter V The Teaching Of Geometry From The Year 1525 To The Present Time

Some Early Printed Books on Geometry 55-59

Paciuolo. Fineus. Instruments employed in field work. Bartoli. Cataneo. Belli. Gargiolli. Gloriosus. The viii Contents

influence of Albrecht Diirer. Geometry of a single opening of the compasses Tendencies in the above works that have a bearing on

teaching 59-60

Germany 60-73

The early universities 60-61

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. Schroter and Gruvius Geometry in the secondary schools of the eighteenth century 65-71 Importance of mathematics and science. The texts of Wolf, Kastner, and Sturm, and their influence. The Realschulen Geometry in the secondary schools of the nineteenth century 71-72 Reorganization of the Gymnasia of Prussia. Admission to universities. General nature of the teaching of geometry and mathematics in general General resum^ from the sixteenth century, showing the development of the teaching of geometry from the standpoint of the curriculum and of method. Tendencies most

prominent 72-73

France 73-87

General survey from the sixteenth century to the middle of the

eighteenth 73-74

Early books that influenced teaching 74-78

Bouvelles. Ramus. Errard. Mercator. Arnauld. he Clerc. The interest in the teaching of geometry up to the time of the expulsion of the Jesuits

The eighteenth century 78-81

General survey. Use of texts in class work. Books of Rivard, Clairaut, and La Caille. Features with educational significance. -The military schools and texts used in them

Legendre and his influence 81-82

The teaching of geom,etry in France since Legendre . . . 83-86 Founding of the government schools. The lyc^es. Suggestions on teaching by Lacroix. By Busset

England 87-90

In the universities 87-88

The thirteenth century. The fifteenth century. First English translation of Euclid. Professorships of mathematics at Oxford and Cambridge. Euclid's " Elements " edited by Barrow, Whiston, Simson, Gregory. In later times by Playfair and Todhunter. The traditions Euclidean Transition of geometry into the secondary schools . , . 88-89

Russia 90-93

Before the creation of the Gymnasia 90

The beginning practical. Influence of Peter the Great in establishing special schools where mathematics was taught. Methods employed in these

Founding of the Gymnasia 91-92

Aim to develop pure science. Introduction of text-books in geometry. The ecclesiastical schools. Attempts to break away from the dogmatic method. Gymnasia begin to send students to the university.

Contents ix

Organization of public schools. Schools for training teachers. Gymnasia programs in the early part of the nineteenth century

Hoi^LAND 93-95

The beginning practical. Books for academic teaching. Beginning of university requirements in mathematics. Geometry in the Gymnasia

Other European Countries 95-96

Austria 95

Influence of the Jesuits. The teaching of geometry in the Gymnasia

Bulgaria 96

Switzerland 96

The United States 96-100

In the universities 96-98

The geometry taught at Harvard, Yale, and other universities up to the time of entrance requirements

In other schools 98-99

The early academies. Colonial grammar schools. First course of the Boston English High School in 1821. The accrediting system

Influence of the English and the French 99

Some references to method .99-100

Summary of the teaching of geometry since the beginning of the

sixteenth century 100-103

Chapter VI Present-day Teaching Of Geometry

Germany 104-106

The place of geometry in the curriculum . . 104-105

The treatment of subject-matter 105-106

Class-room methods 106

The place of geometry in the curriculum . . . 106-108 The treatment of subject-matter . . . . . 108-109

Class-room methods 109

Recent reforms in the teaching of geometry . . 109-112

The place of geometry in the curriculum . 112-115

Recent reforms in the teaching of geometry . . 115-116

The teaching of geometry in the elementary schools 116-117 The teaching of geometry in the two types of normal

schools 117-118

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 affecting the teaching of geometry

Schools in which geometry is taught .... 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

Contents

Other European Countries 128-129

Sweden. Norway. Denmark. Austria. Bulgaria. Belgium. Spain

The United States 130-132

The place of geometry in the curriculum .... 130

The teaching of geometry in the high school . . 130-131

Tendencies. Relation of the high school to the

university. Effects of the "Perry Movement". . 131-132