more knowledge." In the discussion following the reading of Professor Perry's paper several speakers referred to the evils of the present examination system. In reply, Professor Perry heartily concurs and adds, "I assert that we want reform of a whole system of dunce manufacture of which examination is only one part."2

3

Two things above all stand out in Professor Perry's recommendations: (1) That much of elementary geometry be assumed as axiomatic, and (2) that the subject-matter be taught with reference to its utility. We get a better idea of his suggested reforms by examining his syllabus, which has been prepared for use in the training colleges. Under the topic mensuration is included experimental work, such as testing the rules for the areas of the ordinary plane figures, the ellipse, the surface of a cone, of a cylinder, by means of scales and squared paper. Propositions in Euclid are also tested in the same way. Professor Perry recommends as a course in geometry: "Dividing lines into parts in given proportions, and other experimental illustrations of the sixth book of Euclid. Measurement of angles in degrees and radians. The definitions of sine, cosine, and tangent of an angle; determination of their values by graphical methods; setting out of angles by means of a protractor when they are given in degrees and radians, also when the value of the sine, cosine, or tangent is given. Use of tables of sines, cosines, and tangents. The solution of a right-angled triangle by calculation and by drawing to scale. The construction of any triangle from given data; determination of the area of a triangle. The more important propositions of Euclid may be illustrated by actual drawing; if the proposition is about angles, these may be measured by means of a protractor; or if it refers to the equality of lines, areas or ratios, lengths may be measured by a scale, and the necessary calculations made arithmetically. This combination of drawing and arithmetical calculation may be freely used to illustrate the truth of a proposition. A good teacher will occasionally introduce demonstrative proof as well as mere measure1 Perry, op. cit., pp. 12, 13.

2 Ibid., p. 93.

3 Ibid., pp. 25-32.

These train teachers for the elementary schools

ment." Of those who took part in the discussion at the Glasgow meeting the majority agreed with Professor Perry in his attempted reform. The opinion of W. D. Eggar of Eton College shows the attitude of a representative of one of the great Public Schools. He thinks the syllabus excellent, but says there would be great difficulty in adapting it on account of a lack of fully qualified teachers. Mr. Eggar suggests the following treatment of geometry: "A course of geometrical drawing to be agreed upon which should replace Euclid II, IV, and VI. Books I and III to be taught still, but in conjunction with geometric drawing. Euclid II (12, 13) to be transferred to trigonometry, which should begin at an earlier stage." Professor Perry's reform has caused much discussion in England and America. Many English teachers of mathematics are heartily in favor of a change, but they recognize the difficulties in the way, the greatest of which is the rigid examination system controlled by the universities.

The "Perry Movement" will be referred to again in another connection.

OTHER EUROPEAN COUNTRIES

3

The Gymnasia of Sweden were established in 1623 and trained almost exclusively for the clergy and civil offices. These schools to-day are offering the classical and modern courses, covering nine years' work. Mathematics receives its full share of attention. Geometry is usually begun in the second year class and continues with the rest of the mathematics through six years. More attention seems to be given to Euclid than in Germany, but great freedom is allowed the teachers in a choice of texts.*

1 Some of the recommendations suggested by Professor Perry are by no means new. In an article in the Quarterly Journal of Education (London), 1833, it is advised that there should be a preparation for scientific geometry by means of discovering truths from accurately drawn figures. The propositions of Euclid are to be tested by means of instruments and by means of paper cutting. See Quarterly Journal of Education, 1833, Vol VI, pp. 35-49; 237-251.

2 Perry, op. cit., p. 81.

3 Sweden, its People and its Industry, edited by Gustav Sundbarg, containing an article on secondary education, pp. 33-49.

See the Redogörelse for each of the secondary schools at Malmo, Lund, Linköping, Kalmar, Kristianstad, and Karlstad; 1894-95.

The present organization of the Gymnasia of Norway1 dates from 1869, when the literary and scientific courses of six years each were instituted. The middle schools prepare for the Gymnasia and for practical life. Geometry is begun in the middle schools in a practical way and is continued in the Gymnasia, where it is taught in a systematic manner.

2

The Latin and the Realskoler of Denmark, which received their present organization in 1850 and 1871, correspond to the Gymnasia and the Realschulen of Germany. Geometry is taught throughout the six years' course of the Latin schools, where the pupils are prepared for the higher professions and for the university.

Although the teaching of mathematics has had a late development in the secondary schools of Austria,3 since 1884 this subject has ranked equally with the languages and history. The institutions which teach elementary mathematics correspond to those of Germany.

Bulgaria also has the two main types of secondary schools, the Gymnasium and the Realgymnasium. In the four higher classes of each, both algebra and geometry are taught.*

One gains a good idea of the teaching of mathematics in the secondary schools of Switzerland after understanding the aims and methods of the German institutions.

Belgium also has its secondary institutions similar to those of Germany. Mathematics ranks equally with the other subjects. The secondary schools of Spain offer three courses, each extending over five years. Practical geometry and arithmetic are taught in the first year. The teaching of geometry becomes more systematic in the third year, when trigonometry is introduced. Algebra is begun in the second year.

1 Baumeister, op. cit., I2, article on Norway, pp. 399-401; Hippeau, L'instruction publique dans les états du nord, pp. 180-183.

'Baumeister, op. cit., I2, article on Denmark, pp. 385-388; Thornton, Recent Educational Progress in Denmark, in Special Reports on Educational Subjects (London), 1896-97, pp. 587-614; Hippeau, L'instruction publique dans les états du nord, pp. 233-244.

3 Beer und Hochegger, op. cit., Vol. I, pp. 266-532; Simon, Ens. Math., 1902, pp. 157-166.

4

1 Rein, op. cit., I, article on Bulgaria, p. 825.

5 Ibid., article on Belgium, pp. 464-466.

Baumeister, op. cit., I2, article on Spain, pp. 725, 727.

THE UNITED STATES

In the United States, elementary geometry is taught principally in the high schools. It is also found in the courses of the normal schools and the few academies. In the normal schools, the work varies greatly. Some give courses in plane geometry, others in both plane and solid, and still others require plane geometry as a requirement for admission. There are also various kinds of polytechnic schools where geometry is taught more or less with respect to its applications. Our grammar schools now generally give some attention to practical geometry either in connection with mensuration as a branch of arithmetic or as a separate course, where the pupil is taught simple constructions and learns the properties of the common geometric figures.1

Instead of having separate institutions for the various courses, as is the case in Germany, the American high school offers from two to four courses. The Committee of Ten2 (1892) recommended the following four programs: Classical, Latinscientific, Modern Language, and English. In all these, geometry begins in the second year of the four years' course. Algebra is begun the first year, when the pupil is about fourteen years of age. Both studies continue through the third year, when geometry is completed, including both plane and solid. In the fourth year trigonometry and higher algebra are required in the English course. In the other courses there is a choice between trigonometry and higher algebra or history. Although the work of the Committee of Ten has influenced the curricula of the high schools, it has by no means brought about uniformity of courses, a situation perhaps not desirable in this country. The recommendations of the committee, however, will give one a fair idea of the curriculum of the average American high school.

Above all, the high school geometry is logical, although Euclid as such finds little favor with us. The work is generally begun in the second year. Two years earlier, in the elementary school, the student has had some form of concrete geometry. There is no bridging over this gap, it being only the occasional teacher who prefaces the high-school work by some form of inventional geometry.

1 This was recommended by the Committee of Ten, appointed by the National Educational Association. See United States Bureau of Education; Report of the Committee on Secondary School Studies, 1892, p. 23ff.

2 Op. cit., p. 23ff.

The conduct of the class is quite different from that in Germany or in France. In Germany, for instance, the class method predominates. In the United States, the individual is the center of interest. This is seen in the arrangement of blackboards. One black-board suffices for the German or French schoolroom where the class as a unit develops a particular theorem. With us it is necessary to have a continuous board around the room, so that each pupil can draw his own particular figure from which to explain. The above typifies what has been, and is yet to a large extent, our common schoolroom practice. Nevertheless we have teachers who follow the German plan and still others who try to combine these two extremes. The dependence upon a text-book and the requirement of a considerable amount of home work prevent the class period from being a time for instruction. Hence our teachers in mathematics tend too much to "hear" recitations. We have much to learn from Germany and France in this respect.1

Since 1871, when the University of Michigan established the system of accrediting high schools, the teaching of mathematics in the latter schools has been more largely influenced by the universities. We find teachers, therefore, more busily engaged in preparing students for the university than for their daily living. The large universities of the east have not generally adopted the accrediting system, several admitting only by examination. Hence there have been varying standards of admission.. Uniform entrance requirements being desirable, the College Entrance Examination Board, made up of representatives of the various universities concerned, was organized in 1900. This will undoubtedly secure greater uniformity of work in the various preparatory schools and will also improve the teaching of geometry from the point of view of the university, but it does not necessarily follow that this will be best for the pupils.

Within the last few years a movement of another sort has gained some prominence in America. Professor Moore of the University of Chicago was the first prominent mathematician in this country to champion the cause of the "Perry Movement."

1 Chapter VII below contains much that pertains to the teaching of geometry in our schools.

2 See above, pp. 126-128.