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influence of Albrecht Dürer. 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

Schröter and Gruvius

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

Realschulen

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
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.

Le 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 geometry in France since Legendre

83-86
Founding of the government schools. The lycées.

Suggestions on teaching by Lacroix. By Busset
Summary

86-87

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
Summary .

89-90

RUSSIA

90-93

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

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.

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90

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

the nineteenth century

Summary :

92-93

HOLLAND

93–95

The beginning practical. Books for academic teach-

ing. 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. Coloniai

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 :

100

Summary of the teaching of geometry since the beginning of the
sixteenth century

100-103

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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
154-155

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

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