was organized in 1870. About this time (1872) the systematic organization of schools for the training of teachers for the elementary schools was brought about. Two kinds of normal schools, which already existed, have furnished the necessary discipline. These are the Seminaries for Teachers and the Institutes for Teachers, the first of which prepare teachers for the city schools, the second for the schools of the villages. The latter schools are of lower standard than those of the cities. Further advance in public education was made in 1872 by the organization of the Realschulen and Progymnasia. In 1874 the primary schools were reorganized, and in 1889 inferior and secondary technical and trade schools were established. Geometry finds a place in all these institutions.

In the primary schools, practical geometry is studied only in the upper classes of the schools of the second class.1 Here are taught the fundamental properties of plane figures, surveying, and linear drawing. Some of these schools extend their work to the practical study of the familiar solids. The inductive method predominates.

In the intermediate schools, the course extends through six years. Observational geometry is begun in the third year by the study of solids, little attention being given to plane figures. In the fourth year the study of the previous year is treated more in detail and the whole of plane geometry is covered in a general fashion. The practical work is emphasized, finding an application in surveying. In the fifth and sixth years, plane geometry is more fully developed. As for the logical development in these schools there are no demonstrations in the third year, but such work follows directly in the next. As compared with the work in the secondary schools, there is less theory and more practice.

The Seminary for Teachers in its three years' course covers plane and solid geometry with applications to surveying. The Institute for Teachers covers the same subject-matter, but accomplishes this in the first two years of its three years' course. In the former school the work of the first year includes a study of the geometry of the simpler plane figures, and surveying. In the second year, proportional lines, similar figures, regular poly

1 Those of the first class have a course of three years. Those of the second class, four or five years.

gons, measure of areas, and surveying. In the third year, straight lines and planes in space, polyhedra, measure of surfaces and of volumes, and surveying. During this year methods of teaching geometry in the elementary schools are emphasized.

Concerning the methods employed in these schools, Bobynin refers to two "notes" on methodology: "1. The systematic course is preceded by an examination of geometric solids. This is for those who have had no geometric training, but it is also a benefit for the majority in giving them models for future lessons in the school. 2. A new theorem is attacked by the method of analysis, participated in by the whole class. But when a theorem already studied in the class is demonstrated, synthesis only is used, as the analytic method is more difficult for pupils in their early work." Analysis, however, is very little used in the school work.

In the public secondary schools for boys, geometry is begun in the fourth class.1 It is taught five times a week in the Realschule, and twice a week in the Gymnasium and Progymnasium.

The Realschule accomplishes more work than the other schools, the additional subject-matter including proportion and the similarity of triangles and polygons. In the fifth class2 of the several schools, plane geometry is completed and solid geometry is begun. Solid geometry is completed in the sixth year, and in the Realschule all of geometry is reviewed. In the seventh year no geometry is taught, but in the eighth year of the Gymnasium a general review of the previous work in mathematics is prescribed. Algebra is begun in the third year, a year before geometry, and continues through the seventh year. Trigonometry is begun in the sixth year and extends for two years. This sequence differs from that in the German schools. In the latter, geometry is begun a year before algebra, and trigonometry is introduced early enough to be of service in plane geometry.

We thus see that with the reorganizing of the Russian school system on modern lines, geometry receives its full share of attention. In the primary and intermediate schools, practical geometry alone is taught. In the normal schools, where teachers

1 In Russia, the system of numbering the years is like that in America. The highest class is the eighth.

2 Four lessons per week in the Realgymnasium and two per week in both the Gymnasium and Progymnasium.

are trained for the primary schools, demonstrative geometry is taught, but the practical aspects of the subject are ever kept to the front. In the secondary schools for boys, the work covered corresponds to that in the best American high schools.

It has taken the Russians a long time to drive the dogmatic method of teaching from the schools. It is only in recent years that the demonstrative method has predominated. Even now the method of analysis is not sufficiently recognized.

HOLLAND

The secondary schools of Holland1 were organized in 1663. They are of two kinds,2 one class of schools offering a course of five years and the other one of three. The secondary school of five years is the principal institution for the teaching of elementary mathematics. It has a double aim, being a finishing school and at the same time preparatory. It prepares students particularly for the polytechnic, and as a consequence the work in geometry is very thorough. Trigonometry is introduced before solid geometry is begun. The secondary school of three years is not a preparatory institution, and hence the mathematics is more concise and less profound in details. The above schools do not include the Gymnasia. The latter appear originally to have been final schools, but since the creation of the so-called secondary schools, the Gymnasia have lacked more and more the character of final schools in order to assume that of preparatory schools for the university. We are to understand then that the "secondary schools" of five years prepare for the polytechnic and the Gymnasia for the university. Naturally the geometry in the Gymnasia would not be so practical as in the other class of schools. Two courses are offered in the Gymnasia, the literary and the scientific. Cardinaal says,

1 Cardinaal, L'enseignement mathématique en Hollande, Ens. Math., 1900, pp. 307-309.

For the general features of education in Holland, see the article on Holland in Baumeister, op. cit., I2, pp. 671-672.

2 There are two classes of primary schools, those fitting for the secondary schools and those functioning as final schools. By a law of 1878, practical geometry was prescribed for the latter schools, but in 1889 the work was. discontinued and free-hand drawing was substituted.

"In examining the second course and comparing it with that of the secondary school, we see that descriptive geometry is replaced by spherical trigonometry and the first elements of analytic geometry, changes explainable when one thinks of the difference between the studies at the university and those at the polytechnic." We see that the course in advanced mathematics in the Gymnasia is not unlike that in the German Gymnasia, while the so-called secondary schools in Holland seem to be more like the German Oberrealschulen, only the work is not so extensive.

The teaching of geometry on the whole appears to be associated with the preparation for advanced work in mathematics. In the secondary school of three years alone is it taught without reference to such a preparation.

ENGLAND

We have seen that elementary geometry began to be taught in the secondary schools of England in the early part of the nineteenth century, but not in any general way much before the middle of that century. One type of these schools is the great Public Schools, like Eton and Rugby. Another type is the Grammar Schools. The latter must not be confounded with the higher grades of the American elementary schools, for they differ little from the Public Schools in the character of their work. The Public Schools draw their patronage from the wealthy and aristocratic classes, while the Grammar Schools draw theirs from the middle classes.' The English secondary school bears no relation to the elementary school in the sense that obtains in the United States. In the elementary school, children enter at five years and finish at the age of fourteen. The curriculum is shaped with the idea that school life ends then. The secondary school receives boys at the age of seven or eight and graduates them at nineteen prepared either for the university or else for a profession or business life. Since 1837 preparatory schools have fitted boys for the higher classes of the Public and Grammar Schools and for the Royal Navy. They keep boys to the age

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of fourteen, but no longer.1 Geometry is taught in the upper classes of these schools as well as in the secondary schools themselves. We are concerned also with another class of secondary schools, the technical institutions. The University of London, which was founded in 1836, was in part an outgrowth of a demand for technical and industrial education. Various technical and industrial schools were also created about this time, some of which have been preparatory to the University of London and the other modern universities since founded, while other technical schools of a high rank are affiliated with the universities and offer certain equivalent courses. Some of the technical schools are classed with the Grammar Schools. In fact, it is hard to draw any fast line between the various kinds of secondary schools in England. Geometry of a practical nature is also taught in the upper classes of some of the elementary schools. In the higher elementary schools, where the pupil must be not younger than twelve, and remains three years, more serious attention is given to the subject.

4

We now turn to the teaching of geometry in the abovementioned schools. It appears that in some of the higher elementary schools both Euclid and "geometry" are taught, the latter term, as the syllabi of instruction indicate, referring to practical geometry. In one of the typical London schools3 the two following parallel courses are given: In the first year, easy problems on plane figures. In the second year, Euclid I (propositions 1-15), with easy deductions; definitions, postulates, and axioms. What is called "geometry" is taught at a separate hour, it pertaining to the measurement of the triangle, circle, and regular polygons. In the third year, Euclid I with easy deductions. Also at a separate hour elementary "geometry" of the plane, and of the ordinary solids "in easy positions," and simple sections. We thus see that the "geometry" work is becoming a course in mechanical drawing. In the fourth year (course 1 Cotterill, Preparatory Schools for Boys. In Special Reports on Educational Subjects, Vol. 6, 1900, p. 1.

2 The "Public Schools Year Book" gives in its list schools that offer quite technical courses.

3 Medburn Street Elementary School for Boys, Syllabus of Instruction, 1903-04.

4 Algebra, arithmetic, and mensuration are also given this year.

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Trigonometry is also begun as a separate subject.