Syllabus of Mathematics for the Austrian Gymnasien. (Educational Pamphlets, No. 22.) London, Board of Education, Eyre and Spottiswoode, 1910, 10 pp.

E. CZUBER, Der mathematische Unterricht an den technischen Hochschulen. (I. M. U. K., Heft 5.) Wien, Hölder, 1910, 6+39 pp.

R v STERNECK, Der mathematische Unterricht an den Universitäten. (I. M. U. K., Heft 7.) Wien, Hölder, 1911, 6+50 pp.

T. KONRATH, Der mathematische Unterricht an den Bildungsanstalten für Lehrer und Lehrerinnen. (I. M. U. K., Heft 2.) Wien, Hölder, 1910, 27 pp.

J. LOOS, Die praktische Vorbildung für das höhere Lehramt in Österreich. (I. M. U. K., Heft 4.) Wien, Hölder, 1910, 21 pp.

A. HÖFLER, Die neuesten Einrichtungen in Österreich für die Vorbildung der Mittelschullehrer in Mathematik, Philosophie und Pädagogik. (I. M. U. K., Heft 12.) Wien, Hölder, 1912, 103 pp.

Enzyklopädisches Handbuch der Erziehungskunde unter Mitwirkung von Gelehrten und Schulmännern... herausgegeben von J. Loos. 2 Bände. Wien und Leipzig, A. Picklers Witwe, 1906-08.

Articles: "Probejahr," "Hospitieren," and "Pädagogische Seminare."

W. FRIES, Die wissenschaftliche und praktische Vorbildung für das höhere Lehramt. 2. umgearbeitete Auflage. München, Beck, 1910, 6+216 pp.

Band II, Abteilung 1 von Handbuch der Erziehungs- und Unterrichtslehre für höheren Schulen ... herausgegeben von A. Baumeister.

E. MARTINAK, "Zur pädagogischen Vorbildung für das Lehramt am Mittelschulen." Zeitschrift für die österreichischen Gymnasien, 1904, 11. Heft.

A. STITZ, “Die Ausbildung der Mittelschullehramtskandidaten." Mittelschule, 1910, 4. Heft.

III. BELGIUM.

The area of Belgium is less than that of the States of Maryland and Delaware together, but the population is somewhat greater than that of the Dominion of Canada.

Education is controlled by the minister of sciences and arts, who has under him two general directors, one for primary and one for secondary and higher education. For secondary education the ministry also has an inspector general, nominated by the King, and two ordinary inspectors, one for the humanities, the other for mathematics and science. Authority is exercised over schools by the ministry in effective manner through control of the Government appropriations, appointment of teachers, regulation of programs, and prescription of textbooks.

SECONDARY SCHOOLS.

In Belgium the better secondary schools proper may be roughly divided into two classes, those supported by the Government and those maintained by the communes. The former are of two kinds: (a) the Athénées Royaux (royal athenaeums, called also higher middle schools); and (b) the Lower Middle Schools or Middle Schools. The communal secondary schools (collèges communaux) are mostly controlled by the church or religious orders. In 1912 they included 15 collèges, which ranked about as high as the athénées.

(a) The athénées royaux, 20 in number, are subject to official control under the direction of the King. In accordance with a decree of 1888 the courses in the athénées were arranged in three parallel divisions: (1) The humanités grecques-latines, with seven years of Latin and five years of Greek; (2) the humanités latines, with seven years of Latin, no Greek, and a very extensive course in mathematics; (3) the humanités modernes, where modern languages serve as the basis for teaching during the seven years. The three higher classes of the humanités modernes comprise two sections, the scientific section and the commercial section. The classes during the seven years of each of the divisions are numbered VII-I. Pupils entering VII are about 12 years of age and have had the equivalent of 6 years of training in the primary schools.

2

Note that the scheme is somewhat similar to that of the French lycées. In Germany these different types of instruction are given in different schools: the Gymnasium, the Realgymnasium, and the Realschule. 2 The minimum age of admission to the athénées is 11 years, and an entrance examination must be passed.

The mathematical subjects taught in the athénées are arithmetic, algebra, plane and solid geometry, plane and spherical trigonometry, plane and analytic geometry, descriptive geometry, and surveying.

Number of hours per week devoted to mathematics and its applications in the different

divisions.

1 The courses in ancient humanities bifurcate at the beginning of the third year, V.

In the first two years the mathematics is the same for all, and the programs for the Latin humanities and the scientific section of the modern humanities are identical. The commercial section differs from the scientific section in Classes III-I only. In Class III of the scientific section the following subjects are taught:

In arithmetic: General theory of divisibility of numbers, highest common divisor, least common multiple, theory of prime numbers, Fermat's Theorem, conversion of ordinary fractions into decimal fractions and reciprocally, numerical approximations, weights and measures, operations on complex numbers, cube root. In algebra: Discussion of the general equation of the first degree in one and two unknowns, complete discussion of the general equation of the second degree, properties of trinomials of the second degree, questions of maxima and minima, progressions, logarithms, use of tables, compound interest, and annuities. In geometry: Regular polygons, measure of the circle, determination of π, problems, notions on the theory of transversals. In plane trigonometry: Fundamental formulæ, identities, construction, and usage of trigonometric tables, solution of triangles. Surveying and leveling.

In Class I of the Latin humanities two of the eight hours a week are devoted to a thorough review of algebra, geometry, and trigonometry, with new applications of the theories. In the remaining six hours some of the topics taken up are:

Determinants of the second, third, and higher orders; elementary properties, application to the solution of a system of n equations of the first degree; in spherical trigonometry: Solution of triangles, spherical excess, radii of inscribed and circumscribed circles of a triangle, distance between two points on the earth's surface, volume of the parallelepiped and tetrahedron in terms of the edges and angleз; in analytic geometry: Principal properties of points, lines, conics, conics as sections of a cone, intersection and similitude of two conics; in descriptive geometry: Introductory notions.

It may be added that the instruction in the athénée is maintained at a very high standard and carried out in such way as to arouse keen competition for honors and prizes. These are distributed as the outcome of three examinations each year for each class, the third being called the Concours général. A sample paper in the Concours général of 1910 for Class I in Latin humanities and scientific

section will give a further indication of the mathematical standards of the athénée:

I. Analytic geometry.'-Given a rhombus ABCD whose diagonals AC, BD, are, respectively, equal to 2a and 2b, and intersect in 0. (a) Form the general equation of the conics S whose conjugate diameters have the directions of the sides BA, BC, and which meet the diagonal AC in two points E and F such that, OEXOF=-a2. Show that the conics S pass through four fixed points and construct these points. (b) Find the locus of the poles of AC with respect to the conics S. Find also the locus of the points of contact of the tangents drawn to the same conics, parallel to AC. (c) Consider, in each of the conics S, the axes of symmetry I, the polar p of the vertex D, and the perpendicular d dropped from D on p. Find the locus of the points of section of d with I and construct this locus. II. Descriptive geometry.—Given a line c, and the horizontal line a cutting the frontal line b in the point A. (a) Find on the line c the point S, equally distant from the sides of the acute angle formed by a and b. (b) From S drop the perpendiculars SD, SB on a and b, respectively. (c) Construct the quadrangular pyramid S-ABCD, which is found by cutting with the plane (a, b) the solid angle whose four edges are SA, SB, c, SD. (d) Give a representation of this pyramid, applying the conventions with respect to the parts of the drawing of projections of parts seen and hidden. III. Demonstrate that the six dihedrals of a tetrahedron satisfy the relation,

where the four faces are denoted by a, b, c, d, and (pq) denotes the dihedral angle formed by the faces called p and q.

If the trihedral angle formed by the faces a, b, c is trirectangular, show that the preceding equality reduces to

cos (ad)+cos(bd)+cos2(cd)=1.

The student who completes any one of the courses of instruction in an athénée and passes the final examination receives a diplôme de sortie, which admits him to the goal of his ambition, a university. It will admit him to any faculty. Only in the special schools must an applicant for admission, whether he has a diplôme or not, be examined on the program of the Latin or scientific sections.

(b) The State lower middle schools, of which there are about 90 for boys," "were created by the Government to meet the needs of the higher artisan and commercial classes." Entering pupils for these schools and for the athénées have the same preparation. The courses of study are arranged so as to occupy three years. The obligatory courses are: French, Flemish, history, geography, arithmetic, algebra, geometry, zoology, botany, physics, chemistry, commerce, drawing, and gymnastics. As to mathematics, it corresponds approximately to what is taken up in the first four years at the

1 For solution of these questions, see Mathesis, 1911, vol. 31, pp. 35-38, 61, 67.

Mr. Rose seems to be incorrect in stating (p. 351) that there are only about 50 of these schools. Cf. Statesman's Year-Book, 1917, and Reports of the U. S. Commissioner of Education, 1913–14, etc.

athénées. Pupils who have completed the course of a lower middle school are admitted to IV of an athénée without examination or to III after successfully passing an examination.

THE UNIVERSITIES.

There are no higher normal schools in Belgium, and except in very rare instances a candidate for a professorship in an athénée must have received the degree of doctor from a university.

There are four universities-two belonging to the State, at Ghent and Liege; the free university at Brussels; and the largest of all, the Roman Catholic university at Louvain. Each of these universities has certain special schools or institutes connected with it. Perhaps the most famous of them is the technical school attached to the University of Liege. In each of the universities there are four faculties philosophy, law, medicine, and sciences. It is in connection with the last-named faculty that the future professors of mathematics and professors of natural science are formed. On entering the faculty of science as students these candidates are required to present a diplôme de sortie from an athénée or a collège, or else to pass equivalent examinations either (1) before the faculty or (2) before a jury composed of professors of secondary teaching and appointed by the minister of sciences and arts. The students are usually graduates from the scientific section of an athénée.3

In addition to pure mathematics the future professor is required to study general physics and mathematical physics, rational mechanics, chemistry, and crystallography. The program also includes a course in psychology, logic, and moral philosophy, as well as in the history of mathematics.

· Didactic preparation takes place at the same time as scientific preparation, each university possessing a special chair of mathematical methodology.

The scientific preparation extends over four years. During the first two years the student prepares to secure the certificate as candidate in physical sciences and mathematics. For three years the courses are the same for all the students of mathematics; in the fourth year each takes up, according to his tastes, one or other of these groups: Analysis (including differential geometry), higher geometry, astronomy and geodesy, rational mechanics and celestial mechanics, physics. The thesis for the doctorate is on a question related to the group chosen.

1 Selected titles from the official list of ma the matical texts used inthe athénées and lower middle schools are given on pages 234-235 of the report of the subcommission.

The buildings of this university were completely destroyed by the Germans Aug. 26, 1914.

To meritorious and needy students the State awards, on the basis of a concours, annual scholarships amounting to 400 francs. These scholarships may be received each year of the course. There is generally one such scholarship for the section of mathematics in each university.

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