QUEENSLAND.

Queensland does not yet possess any special college for the training of teachers. In the "grammar schools" the staff consists principally of university graduates. The high schools also aim at making a university degree an almost indispensable qualification for their

teachers.

The University of Queensland, which was founded at Brisbane in 1909 and formally opened in 1911, has an important sphere of influence, and the same general scheme of mathematical instruction as in other States of the Commonwealth is in vogue.

TASMANIA.

Under the scheme for training of teachers which is being introduced in Tasmania candidates pass the senior public examination and matriculate at the university. The next year is spent in teaching in selected schools. Thereafter they enter the training college at Hobart, the minimum age at entrance being 18 years. They attend university classes for two or three years. All receive professional training at the college.

Mathematical courses at the university are similar to those already described.

WESTERN AUSTRALIA.

The organization here being in its infancy, there is nothing of exceptional interest to record.

The universities of Australia are staffed by British professors, and thus the mathematical work of the country is fashioned in conformity with much the same ideals as those held in the motherland.

BIBLIOGRAPHY.

H. S. CARSLAW, The Teaching of Mathematics in Australia. Report presented to the International Commission on the Teaching of Mathematics. Sydney, Angus & Robertson, 1914. 79 pp.

In the above sketch I have many times quoted verbatim from Prof. Carslaw's most readable report J. B. TRIVETT, The Official Year-Book of New South Wales for 1915. Sydney, W. A. Gulick, 1917.

"Education," pp. 213-274.

Rept. of U. S. Commis. Educ., 1915-16, Vol. 1, Washington, 1916.

"Education in Australia," pp. 641-659.

Two other recent publications contain considerable information about the universities:

(1) Universities in the Overseas Dominions, Board of Education, Special Reports on Educational Subjects, vol. 25, London, 1912, pp. 116-171, 196-198, 238-269;

(2) Congress of the Universities of the British Empire, 1912, Report of the Proceedings, London, 1912; also the Year Book, 1914.

II. AUSTRIA.

Austria is nearly 116,000 square miles in extent, and its population totals in the vicinity of 29,000,000. Of these, about 10,000,000 are Germans; nearly 6,500,000 are Bohemians, Moravians, and Slovaks; at least 5,000,000 are Poles; and some 3,500,000 are Ruthenians.

The minister of public instruction is at the head of the Austrian educational system. He has inspectors of secondary schools in each of the 14 Provinces of the Empire and certain matters of administration are assigned to local boards. Church and private schools are subject to the same regulations as those of the State.

SECONDARY SCHOOLS.

Before 1908 there were three general types of middle or secondary schools: (1) The Gymnasium, (2) the Realschule, and (3) the fouryear Realgymnasium, the courses of study in all of which were based upon four years of work in the primary schools. The first two of these types are still the most prominent. The normal age of the pupil commencing secondary education is about 10 years.

The complete Gymnasium provides for a course of eight years' study, divided into two parts of four years each. The first part is referred to as the Untergymnasium; the latter as the Obergymnasium. The course of study is characterized by the great emphasis laid on instruction in Latin, Greek, and German. In passing from one class to another the pupils undergo very searching examination. This is characteristic of all the Austrian secondary schools.

The seven years' course of the Realschule consists of two cycles of three and four years each, corresponding to programs of the Unterrealschule and Oberrealschule. Here the emphasis is laid on modern languages (Latin and Greek are not taught at all), mathematics, physics, chemistry, etc.; that is, on those subjects which are designed to impart technical knowledge and afford suitable training to those intending to follow industrial pursuits.

As a result of a ministerial decree in 1908, the establishment of Realgymnasien and Reform-realgymnasien, each with an eight-year course, was also authorized. But, in what follows, the Gymnasium and Realschule only will be treated as representative of Austria's best secondary schools.

■ In 1915-16 there were 336 Gymnasien, of various forms, for boys and 148 Realschulen.
101179°-18-

-2

15

The number of hours per week devoted to class work in mathematics (M.), geometric drawing, and descriptive geometry (D. G.), in these types of schools, is exhibited in the following table:

23

10

25

15

The total number of hours in these subjects represents about 14.7 per cent of all class periods in the Gymnasium and about 16.5 per cent of those in the Realschule.

In classes I-VIII of the Gymnasium instruction is given, according to the program of 1909, in arithmetic, algebra, plane and solid geometry, plane trigonometry, plane analytic geometry, and elements of calculus.

To give a definite idea of the nature of the contents of these courses a few selections from the program may be made:

1

Class V: Extension and completion of arithmetic and algebraic studies of the previous class. Continued practice in solution of equations of the first degree arising "from the manifold spheres of thought in which they may be applied." Practice in powers and roots by straightforward examples. Solid geometry.-Oblique projections 1 of the simplest bodies (also of crystalline forms). Plan and elevation of simple objects by observation and common sense. Conceptions and laws concerning the mutual relations of straight lines and planes; limited to the fundamental and typical propositions and proofs with constant appeal to observation and intuition. Properties and calculations for surface and volume of prisms (cylinders), pyramids (cones), the sphere, and their sections and portions when in section. Euler's theorem (S+F=E+2); regular polyhedra. Obergymnasium-Class VI: Trigonometry.-The trigonometric ratios, their geometric representation, especially for the purpose of giving a firm grasp of the characteristics and relations of these functions. Solution of triangles. Repeated comparison of trigonometric propositions and methods with those of plane and solid geometry. Varied applications of trigonometry to problems in surveying, geography, astronomy, etc., the data where possible to be obtained from direct (even though rough) measurements by the pupil. Class VII: Algebra.— Arithmetic progressions (of the first order),2 geometric progressions. Application of the last especially to compound interest. Permutations and combinations in simplest cases. Binomial theorem for positive integral exponent. Theory of probabilities. Analytic geometry.-As a continuation of the graphic representation of single functions previously given, further use is made of the analytic method in dealing with lines of the first and second degree with reference, when occasion offers, to the geometric treatment of the same figures and their relations. Representation of direction coefficients (chiefly those of the curves dealt with in class) by means of derivatives. Simple problems of differentiation in connection with problems in mathematics and physics. Approximate solution of algebraic equations (and of the simplest transcendental equations which occur) by graphic methods. In Class VIII the two hours a week are

1 Drawings of solid bodies to show several faces, etc., not in true perspective, but free-hand or constructed according to very simple rules. Cf. A. Höfler, Didaktik des mathematischen Unterrichts, p. 207. Series such as 12+22+32+ . . .”

2 Ordinary arithmetic progressions are said to be of the first order. 18+23+33 +...., are called A. P. of higher orders.

given up to a comprehensive recapitulation of the whole range of school mathematics, especially equations, series, solid geometry, trigonometry, and analytic geometry. Broader and deeper treatment of particular parts of the subject. Applications to the various subjects of the curriculum and to practical life in place of merely formal exercises. Retrospective and prospective consideration from historical and philosophical points of view.

For the Realschule the work is very similar. There is the same effort to correlate mathematics with other branches of instruction on which it happens to bear and with practical applications in actual life. Though searching examinations of "orientation" and "classification" have been the portion of students of the Gymnasium and Realschule each year, the final examinations (Maturitätsprüfungen) leading to the certificate of maturity were formerly often much feared. Some amelioration with respect to the extent of detail here required was made imperative by the decree of 1908.1

The main object of these examinations is to determine whether or not the maturity, general efficiency, and development of intelligence on the part of the candidate is sufficient to allow him to take up studies in the universities or higher technical schools. Whilst the examinations for "classification" establish in some measure the rating of a pupil's acquisition with reference to a certain part of the program taught, the "Maturitätsprüfung," on the other hand, embraces the whole range of knowledge acquired by the pupil in the Gymnasium or Realschule.

The examination consists of two parts, a written and an oral. The former includes questions on the required languages, on mathematics, on exercises in descriptive geometry, and tests in facility with freehand drawing. The oral examination is on geography, history, mathematics, descriptive geometry, physics, chemistry, and natural history.

The examination commission pronounces its verdict after consideration of all the written and oral examinations as well as of the grades received by the candidate in tests during the last year of his course. When a candidate fails he may present himself a second time at the end of a semester or of a year, but he may not repeat the examination more than twice.

UNIVERSITIES AND PREPARATION OF SECONDARY SCHOOL TEACHERS.

There are eight State universities in Austria. The largest is the University of Vienna (German), with over 10,000 students. The oldest, dating back to the fourteenth century, is the German-Bohe

1 Cf. F. Wallentin, Maturitätsfragen aus der Mathematik. Zum Gebrauche für die obersten Klassen der Gymnasien und Realschulen zusammengestellt. 10. Auflage. Wien, C. Geroldssohn, 1912. 8+197 pp. Auflösungen, 5. Auflage, 1906. 8+221 pp.

*In the summer of 1916 this number was reduced to 3,472

mian University of Prague. There are three other German universities, at Gratz, Innsbruck, and Czernowitz. The Polish universities are at Krakow and Lemberg.

Instruction in the universities is imparted by means of: I, general courses; II, special courses; and III, exercises in proseminary and seminary.

I. The general courses are organized with a double aim: (1) To furnish fundamental theoretical knowledge to those studying mathematics in a purely scientific spirit and proposing to terminate their studies with the doctorate; (2) to prepare students destined for secondary-school teachers. Especially in view of this second aim are the general courses usually organized in cycles of three or four years, and each year a course for beginners is arranged that they may be taught differential and integral calculus as soon as possible, since it is necessary for the study of theoretical physics.

During the five years 1905-6 to 1909-10 professors at (1) the University of Vienna and (2) the German section of the University of Prague gave the following general courses:

(1) Differential and integral calculus, 5 hours weekly during 2 semesters; theory of numbers, 5 hours during 2 semesters; theory of differential equations, 5 hours during 2 semesters; calculus of probabilities, 3 hours; definite integrals and calculus of variations, 5 hours during a semester; theory of linear differential equations, 5 hours; elliptic functions, 5 hours; theory of functions, 5 hours during 2 se resters; algebra, 5 hours during 2 semesters; analytic geometry, 4 hours during 2 semesters; theory of invariants with geometric applications, 2 hours; algebraic curves, 2 hours; curves and surfaces of the third order, 2 hours; synthetic geometry, 4 hours during 2 semesters; differential geometry, 2 hours during 2 semesters; line geometry, 2 hours; continuous groups, 2 hours; noneuclidean geometry, 2 hours; insurance mathematics, 4-6 hours during 2 semesters; mathematical statistics, 3 hours; sickness and accident insurance, 2 hours.

(2) Applications of infinitesimal calculus to geometry, 3-4 hours; theory of invariants, 2 hours; differential and integral calculus, 4-5 hours during 2 semesters; systems of algebraic equations, 1 hour; differential equations, 5 hours; introduction to calculus of variations, 1 hour; fundamental notions of analysis, 2 hours; analytic geometry, 3 hours through 3 semesters; elements of the theory of functions, 3 hours; elements of the theory of numbers, 2 hours; algebraic equations, 4 hours; introduction to descriptive geometry, 2 hours; characteristic features of infinitesimal calculus, 3 hours during 2 semesters; theory of groups and algebraic equations, 2 hours; selected chapters of analytic geometry, 2 hours; vector analysis, 2 hours; differential equations, 3-5 hours; contact transformations, 2 hours; theory of transformations, 5 hours. An equally elaborate