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struments and gains conceptions of the different geometric figures. The work of the following year links very closely with this preparatory work, but the logical now assumes prominence. The course includes the subject-matter of Euclidean geometry, but Euclid as a text has no place in the German schools. In fact, very little dependence is placed upon any text. The value of modern geometry is being recognized, for in the programs mention is made of harmonic points and rays, and the theory of transversals, and algebraic and trigonometric methods are freely applied to the work in geometry. In fact, there is such an interlacing of the work that one sometimes finds it difficult to name the subject that is being taught.1

In the German schools, the classroom is a place for instruction, not a place where the pupils "say" their lessons. The lesson in geometry begins by the teacher calling a boy to the single small blackboard behind the teacher's desk. The figure is drawn, the teacher meanwhile questioning either the boy at the board or the other members of the class. A single student rarely gives an entire proof except in resumé. The teacher is careful that all members of the class participate, the method being that of class instruction as opposed to individual instruction. The work is very thorough, if necessary the whole hour being given to a theorem with its applications and corollaries. Time is not lost, for the students understand the work, and when the foundations are once laid they are secure. Very little homework is assigned, the exception generally being in connection with some construction problem. With teachers who know their subjects, who know how to economize time, and who know how to instruct, the mathematical education of the German boy is well provided for.

FRANCE

The lycées2 are the great institutions for secondary education in France. They correspond to the German institutions just described, but with this difference: In Germany, the courses vary with the different institutions, while in France, as in the United States, one institution gives a choice of several courses.

1 This is particularly true of the upper classes.

2 In the smaller cities are found "collèges," which are also secondary institutions, but the lycées are of higher standard.

The course of study for the lycées and colleges given in the decree of May 31, 1902,1 instituted the new plan of dividing the course of study into two cycles of four and three years each. In the first cycle the pupil has the choice of two courses. In the A course, Latin and Greek are obligatory. In the B course, science and drawing receive greater attention, Latin and Greek not being required. In the second cycle, extending through three years, there is a choice of four courses, namely: A, Latin and Greek; B, Latin and a Modern Language; C, Latin and Science; and D, Modern Languages and Science.

We shall consider first the two courses of the first cycle. In the second year (cinquième) of the scientific course (B),3 the pupils perform geometric constructions by means of the setsquare, the rule, the compasses, and the protractor. They study the properties of triangles, parallelograms, circles, and other plane figures, but strict logical work is not yet begun. In the classe de quatrième, the logical work receives more emphasis. By the end of this, the third year, the main part of the subjectmatter of plane geometry is completed. In connection with the treatment of similar triangles, the definitions of the trigonometric functions are introduced. The construction of simple curves like the cissoid and conchoid is undertaken toward the end of the year. In the fourth year (troisième), solid geometry is studied together with some work in surveying.

The classical course (A) of the first cycle gives a much more limited course in geometry. The work is begun a year later1 than in the scientific course. In the fourth year plane geometry

is finished except for some of the work in mensuration.

In the second cycle of three years, the mathematics is the same in the A and B courses as it is in courses C and D, where it is emphasized the most. In the first year (seconde) of this cycle, in the A and B courses (the classical and literary), solid

1 Plan d'études et programmes d'enseignement dans les lycées et collèges de garçons (Arrêtés du 31 Mai, 1902).

2 As in the Gymnasia of Germany, the classes are numbered opposite to the method in the United States. In the classe de cinquième the pupils are about twelve years of age.

3 The plan of 1902 as modified by the new program for mathematics of September, 1905. See Ens. Math., 1906, pp. 65-77

4 In the third year (quatrième).

geometry is studied. In the second year (première), further work is given in plane geometry involving metrical relations. Here the fundamental notions of trigonometry are introduced and solid geometry is completed. In the third and last year (classe de philosophie), no geometry as such is studied, attention being given to advanced algebra and its applications in analytic geometry.

In the C and D courses of the second cycle, the mathematics is more extensive than in the courses just described. In the first year attention is again given to plane geometry. The previous work is reviewed and expanded. The notion of geometric locus is introduced for the first time and trigonometry is further developed. The practical work of the year ends with applications in simple surveying. In the second year of the second cycle, solid geometry is again studied and completed. Trigonometry for the first time becomes a separate subject. The last year (classe de mathématique) provides for a review of the previous work and introduces phases of modern geometry. Higher studies in mathematics are also begun in this year.

We notice that in the French as in the German schools geometry extends through several years. It is begun in the classe de cinquième in the scientific course, when the boy is eleven or twelve years old, and extends six years, to the completion of the second cycle. Solid geometry and algebra are begun in the classe de troisième and both continue for four years. Trigonometry is begun at first in connection with geometry in the third year of the first cycle in the scientific course and continues to the end, a period of five years. We thus see the same interlacing of the mathematical subjects as in the German schools. As in the German Realgymnasium and Oberrealschule, the mathematical course in the French lycée easily includes the work of the first year of an American college.

Euclid has no place in the French schools. Even the use of Legendre as a text is seen only here and there, the teachers preferring to use texts of their own or of a colleague. The first work in geometry is practical and the transition into the logical is quite gradual, but when the logical standards are once established, the teachers are very exacting with their pupils. When the more scientific study of geometry is begun, the algebraic method is employed whenever opportunity offers. In like man

ner trigonometric methods are used after proportion is reached. The books used do not always show this interlacing of the subThe teachers emphasize this particularly in the exercise

jects. work.

As in the German schools, the class and not the pupil is the unit for instruction. At the first of the hour the teacher returns the note-books, which contain the previously assigned written work (devoirs). If no great difficulties are encountered only a few comments are made on the corrected work. If the assigned work be difficult and the students have not done well, the work is all carefully explained in class. As in the new work, a pupil passes to the one small blackboard, and under the questioning of the teacher, the exercises are explained by the class. This correcting of the written work may take the whole hour. The teacher is very thorough and does not sacrifice quality for quantity. As a rule it takes but a few minutes to correct the assigned work, and then some new exercises are assigned for the next week. Several pupils may pass to the board during the hour. The pupils invariably explain what they are doing while they draw the figures. The figure being drawn, the teacher seeks to bring the whole class into the work. One feature of schoolroom practice, however, tends to suppress the spontaneity of the class. Whether the teacher be explaining, or the pupil reciting, it is the practice of the class to keep taking notes. The result is that quick, sharp work on the part of the pupils is frequently lacking, so that the teacher has to do a great deal of the talking. One good feature of the classroom method is the care of the teacher to bring out always the practical side of the work, but this is limited to the field of mathematics. There are no applications of geometry to field work in surveying or to the other sciences."

A movement in France for the teaching of solid geometry in connection with plane geometry has gained considerable headway

1 This statement, like much of the description of the French schools, is based on personal observation in the lycées of Paris. It is assumed in the above that the work there is typical.

2 Based on data obtained from the lycées in Paris in June, 1905. In the last year this application to science is made after the subject-matter of elementary geometry has been completed. The new course, in effect September, 1905, provides for the applications of geometry to surveying before the course in geometry is completed

2

in recent years. The "Nouveaux éléments de géométrie" by Professor Charles Méray, which appeared in 1873, has, after an interim of over twenty years, claimed the attention of French teachers of mathematics. Méray, however, is not the first to have originated this idea.1 Gergonne, in about 1825, raised the question if it were natural to separate the teaching of solid geometry from that of plane. Before Gergonne, Lacroix in 1816 pointed out the analogies between certain theorems of plane and solid geometry and suggested that it is possible to combine this teaching in certain cases. Mahistre in 1840 pointed out the advantages of employing this principle of analogy in the teaching of geometry.' Valat in 1866 published a pamphlet on the reforms of teaching elementary geometry. In this he gives a syllabus which shows this principle of analogy, and by the sequence of the books one can see the idea of "fusion" fully carried out. Books I, III, V, and VII are on plane geometry, and Books II, IV, VI, and VIII on solid geometry. Book II in solid geometry embodies subject-matter analogous to that in Book I of plane geometry, and so on with the rest. For example, Book III contains in Chapter I, circles, arcs, chords, and measure of arcs. Chapter I of Book IV treats the sphere and its general properties. Chapter II of Book II is on the intersection and contact of circles. Chapter II of Book IV, on the intersection and contact of two spheres. Chapter III of Book III, similar figures. Chapter III of Book IV, similar solids. Chapters IV in both books treat of areas of figures. The author ranges his subject-matter on the page in two parallel columns, the books of plane geometry on the left, those of solid geometry on the right, with the books and chapters in both systems set over against each other. In such a scheme, by studying the odd numbered books, plane geometry could be studied first, or after each book in plane geometry

1It may be interesting to recall that Menelaus proved a theorem in plane geometry as a lemma to the one corresponding in spherical geometry. See above, pp. 36-37.

'Loria, Sur l'enseignement des mathématiques élémentaires en Italie, Ens. Math., 1905, pp. 11-20.

3 See above, pp. 84-85.

4 See Loria, Ens. Math., p. 15; L. Ripert in Ens. Math., 1899, p. 62.

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Valat, Plan d'une géométrie nouvelle ou réforme de l'enseignement de la géométrie élémentaire.

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