of what is taught in secondary schools, and exercises in space perceptions; accuracy and speed in the solution of simple examples applying the idea of a function and the elements of differential and integral calculus to functions coming up in secondary school work.

For (a) descriptive geometry and (b) geometrical drawing: (a) Thorough grounding in orthogonal, oblique, and central methods of projection including axonometry, knowledge of relief perspective, of the most important map projections, especially of stereographic projection and of cyclography. Familiarity with the constructions relating to curved lines (especially curves of the second order, space curves of the third and fourth order and helices) and curved surfaces (chiefly surfaces of the second order, surfaces of revolutions, helicoidal, ruled and envelope surfaces, particularly in lighting-constructions). Acquaintance with some applications of descriptive geometry (as construction of sundials, roof trusses, sections of stone (stereotomy)). Knowledge of projective and infinitesimal geometry in so far as these subjects are necessary in the applications of descriptive geometry. Exactness and facility in constructive drawing.

(b) Elements of descriptive geometry to the extent of the program of Realschulen. Axonometric representations. Elements of shadows and linear perspective; the geometrical construction of, and by means of, polygons and the most important plane curves, especially the conic sections. Exactness and facility in constructive drawing. Each examination comprises three parts: The Hausarbeiten (theses), the written examinations, and the oral examinations.

To prepare each of the theses the candidate has three months. These theses need not necessarily show capacity for discovery on the part of the candidate, but they must indicate thorough familiarity with the literature and content of the subjects. In connection with a very large subject a general presentation of the outstanding results may be permissible. Again, certain illustrations of a theory may be worked out. The character of the topics is shown by the following selection of themes of theses in recent years. (None of these subjects were discussed in the lectures at universities where they were assigned.)

(a) For major:

The theory of Fourier's series.

The isoperimetric problem.

Systematic presentation of the proofs of the fundamental theorem of algebra.
On Abel's equation.

Theta functions and their applications in the theory of surfaces of the fourth
order.

Historical presentation of the progress in the theory of algebraic equations (in its leading features).

Triply orthogonal systems of surfaces.

Transcendental numbers, especially e and л.

Jacobi's functional determinants and their most important applications.
Method of derivation of large prime numbers.

(a) For major-Continued.

The series for tanx and secx and the most important properties of their

coefficients.

The significance of in the calculus of probabilities and theory of errors.
Focal properties of surfaces of the second degree.

Algebraic treatment of the 27 lines on a surface of the third order.

Reye's complex.

Rational space curves of the fourth order.

dz

Study of the representation by w= √ √ (z—a) (2—b) (2—c)

(b) For minor:

3/

Solution of equations of the third and fourth degree.

Properties of the nine-point circle.

Theorems of Pascal and Brianchon and their proof in the case of the circle. The general term of the Lamé series and the proposition with regard to the greatest common divisor of two expressions.1

Elementary geometric treatment of the problem of tangencies of Apollonius. A presentation of pages C2-G of Girard's Invention nouvelle en l'algèbre (1629) in modern mathematical phraseology and notation.

The determination of the 15 Archimedean solids in terms of the radius of the circumscribed sphere.

Weierstrass's theory of irrational numbers.

The theorems of Fermat and Wilson.

In connection with the written and oral examinations, questions are usually asked on the theory of symmetric functions, the algebraic solution of equations of the first to the fourth degrees, and better candidates may also be questioned on the elements of the theory of groups and on the proof of the impossibility of the algebraic solution of equations of degree higher than the fourth. Questions on the calculus of variations are usually omitted. In general, clarity of perception and certainty in handling fundamental theorems are valued more than the extent of minute knowledge.

In the written examination there is also opportunity to indicate ability to apply theoretic knowledge to practical problems. Further, the examinations pay special attention to the subjects of the students' university lectures and seminary exercises. The written examination for a major lasts eight hours, two sessions of four hours on the same day; for a minor four hours are allowed.

In 1908-1910, 241 persons passed the professorship examination, with mathematics and physics as majors; 89 at Vienna, 80 at Prague, at Czernowitz, etc., only 1.

Comparatively few students in Austria proceed to the doctorate in mathematics-less than 50 in the last 45 years. Half of these were at Vienna.

In the examinations of candidates with mathematics as a minorthat is, of those who may expect to become teachers in the lower

This title refers to G. Lamé's "Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers", Comptes Rendus, 1844, tome 19, pp. 867-870. The series of numbers 1, 1, 2, 3, 5, 8, 13, 34, . . ., which here comes up, is the recurring series often called the Fibonacci Series.

mathematical classes of the middle schools-the ground covered is usually that of the Maturitätsprüfung. Some examiners also demand spherical trigonometry and such things as the solution of an equation of the third degree or lay special emphasis on the development of the power of space perception.

It is felt very strongly by some Austrian educators that the lower standard for teachers in the Untergymnasien and Unterrealschulen is decidedly detrimental to the best interests of these institutions. After a candidate for a professorship in a secondary school has passed the professorship examination, he has yet to undergo a “trial year" before he can be assigned to a post. At least this is true in theory; in practice the need for teachers has been so great that most of them have not had adequate professional preparation. Of over 200 candidates approved by the Vienna examination commission and supposed to be having a "trial year" in 1908-9, only one actually completed the work of the year.

This trial year is passed in a Mittelschulseminar, which is most successful when it is directly connected with a Gymnasium or Realschule. On entering the Seminar the student is placed under a certain professor who has charge of his development during the trial year. In the first few weeks the candidate visits classes which his professor teaches, and notes the methods employed and the personality of the pupils. The candidate next gives instruction with the aid or under the direction of the professor. At the beginning of the second semester the candidate instructs without aid for at least a month. Furthermore, all candidates have weekly conferences with the guiding professor and the director of the Seminar with reference to various questions regarding instruction, school discipline, pedagogy, school hygiene, and notable publications in pedagogic literature.

After 8 years of service professors in secondary schools are entitled to a small pension in case of need. The rate of pension gradually increases until after 30 years of service it amounts to full salary.

BIBLIOGRAPHY.'

F. BERGMANN, Der mathematische Unterricht an den Realschulen. K. KRAUS, Der mathematische Unterricht an den Volks- und Bürgerschulen. Berichte über den mathematischen Unterricht in Österreich, veranlasst durch die Internationale mathematische Unterrichtkommission Heft 1. Wien, A. Hölder, 1910, 6+81 pp.

E. DINTZL, Der mathematische Unterricht an den Gymnasien. (I. M. U. K., Heft 3.) Wien, Hölder, 1910, 8+78 pp.

A. ADLER, Der Unterricht in der darstellenden Geometrie an den Realschulen und Realgymnasien. E. MÜLLER, Der Unterricht in der darstellenden Geometrie an den technischen Hochschulen. (I. M. U. K., Heft 9.) Wien, Hölder, 1911,

124 pp.

1 See also L'Enseignement Mathématique, tome 10, 1908, pp. 516-522; tome 12, 1910, pp. 326-341; tome 13, 1911, pp. 159–166, 237-243, 332-33; tome 15, 1913, pp. 79–84, par J. Renard.

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

I 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. ? The minimum age of admission to the athénées is 11 years, and an entrance examination must be passed.