judges, for half an hour each; (c) a practical exercise consisting in the solution of a problem in descriptive geometry or in rational mechanics drawn, by lot, from the problems of the questionnaire, and response to observations of the tribunal.

The examinations for secondary school professorships are open only to licentiates and comprise: (1) A written examination four hours in length, consisting of the development of two themes of a questionnaire; (2) an oral examination consisting of response to five questions drawn by lot; (3) a practical exercise; and, after the exclusion of those who have not met with success, (4) delivery of a lesson after eight hours of preparation to be criticized by one or two opponents; and (5) exposition of a given topic and replies to criticisms of the tribunal.

The best mathematical positions in the secondary schools are obtained by the doctors of mathematical sciences, who are also eligible for positions on the faculty of sciences of a university. To prepare for the doctorate it is necessary to follow courses at the university in higher analysis, advanced parts of geometry, astronomy of the planetary system, and mathematical physics, and to present a memoir on a subject selected by the candidate and satisfactorily sustained against objections on the part of the tribunal.

The titles of some advanced courses offered to those preparing for the doctorate may be given: Ordinary differential equations; calculus of variations; integral equations; quaternions; functions of a complex variable; elliptic functions; Galois's theories.

It usually takes four years in the university to pass the licenciatura, and one extra year, for those who are apt scientific investigators, to make the doctorate.

The professors of secondary and higher education are appointed by the King; in the case of the institutos one of the professors is appointed as director.

BIBLIOGRAPHY.

Z. G. DE GALDEANO, Nueva Contribución á la Enseñanza de la Matemática con indicaciones de sistematizacion matematica. Zaragoza, E. Casañal, 1910. 64 pp. L'enseignement mathématique en Espagne. (Commission internationale de l'enseignement mathématique. Sous-Commission espagnole.) Zaragoza, E. Casañal, 1910. 8 pp. 2éme rapport, Zaragoza, E. Casañal, 1911. 18 pp. C. J. RUEDA L'enseignement des mathématiques en Espagne. Mémoires presentées au congrès de Cambridge par M. C. J. R. délégue en Espagne de la commission internationale de l'enseignement mathématique. Tome Ier Madrid, Tip. de la "Rev. de Arch., Bibl. y Museos," 1912. 143 pp.

Among the memoirs in this volume are: "M. Torroja et l'évolution de la géométrie en Espagne" by M. Vegas; "Enseignement de la géométrie métrique à la Faculté des Sciences" by C. J. Rueda: "Les cours d'analyse mathématique aux Facultés des Sciences espagnoles" by L. O de Toledo: "L'enseignement du calcul infinitésimal aux Facultés des Sciences espagnoles" by P Peñalver; "Enseignement des mathématiques aux Écoles Normales" by L. Ferreras.

XVI. SWEDEN.

The area of Sweden is a little less (excluding the lakes) than 170,000 square miles, and the population was estimated to be on December 31, 1916, about 5,757,000. Of these, all but about 100,000 belong to the established Lutheran Church. It is, therefore, not surprising to find church and school both placed under the administration of the Ecklesiastikdepartementet, or the ecclesiastical department. The trend of circumstances recently has been toward their separation. It was not so long ago that the chapters (domkapiteln), composed of ministers and laymen, were still the local boards of administration, not merely of ecclesiastical affairs, but also of affairs relating to the secondary schools and to the elementary school system. Since 1905, however, the central government of the State secondary schools and equivalent educational establishments receiving State aid has been in the hands of a board called the Royal Board of Secondary Schools; the result has been that the powers of the chapters as regards these educational institutions have been very considerably curtailed. In 1913, by the establishment of a central board for the elementary schools, these also were placed under the administration of expert laymen.

The royal board of secondary schools deals with such matters as curriculum, discipline, training of teachers, appointment of teachers, etc. It is also the duty of its members to inspect the schools personally and to give instruction and advice in the course of their inspection.

Since 1905 the State secondary schools for boys have been classified into two groups: Realskolor or modern schools (independent), and högre allmänna läroverk, each comprising a realskola and a gymnasium.

There are 77 secondary schools for boys (allmänna läroverk), 38 of these are högre allmänna läroverk and 39 are independent realskolor. Among the latter, 18 are coeducational schools. There is no independent State gymnasium.

Into the realskola, which has six one-year classes (one, the lowest, to six), the boy may enter at 9 years of age. The course at the gymnasium is based upon the work of the five lower classes of the

Note that the secondary education is built on the third year of primary education, instead of the fifth, as in Denmark. The primary schools give a six-year course, while in some cases continuation courses are offered for three years longer.

realskola and is divided into two "sides"; the Latingymnasium and the realgymnasium, each consisting of four one-year classes called "rings" (I to IV). The boys in the sixth class of the realskola are thus of the same age as those in the first ring of the gymnasium.

REALSKOLOR.

Mathematical instruction, which occupies about one-sixth of the pupil's time, is here given during five hours weekly in each of the classes except in the first and fifth, where it occupies four hours. The subjects taught are arithmetic, algebra, and geometry. In the sixth class the pupils are instructed in: (1) Algebra-evolution and involution, proportion, equations of the first and second degree with one unknown, graphs, and problems; (2) geometry-geometric exercises and amplification of preceding course, which has dealt with circles and polygons and simple problems. Drawing exercises of the Realskola include, in class 4, geometric construction of parallel lines, triangles, parallelograms, and polygons; in class 5, drawing of regular figures, such as the ellipse and limaçon; elements of descriptive geometry; in class 6 further exercises in descriptive geometry.

In 1904-5 about 60 out of 75 schools used Euclid's Elements as textbook in geometry. Four years later, with the development of the more practical or modernized scheme of secondary education, about 60 out of the 75 schools had adopted texts similar to the school geometries now so common in England.

The final goal of the realskola is a State examination (Realskolexamen), which gives admittance to various technical schools and to schools of forestry, agriculture, and mining, and qualifies for various appointments in the post office, railway or telegraph service, etc. The examination consists of two parts, the one written and the other oral. The questions of the first part (in Swedish, German, English,

and mathematics) are the same for the whole country and the requirements are moderate.

GYMNASIA.

In 1913 about 57 per cent of the pupils in the gymnasia were in attendance at the realgymnasia.

The number of class periods per week in the realgymnasium (including gymnastics, fencing, singing, and religious instruction) is 38 to 41, of 45 minutes' duration. There must be a pause of 10 minutes between two periods. About one-quarter of the total time, apart from instruction in gymnastics, fencing, etc., is given to mathematics and drawing. In order to avoid overpressure and to permit of a pupil's devoting himself to some special study for which he displays marked aptitude, some options (valfrihet) are allowed in the last two years of the gymnasium course.

The extent of requirements in the different rings of the gymnasium may be seen in the following table:

1 Those pupils who have elected Greek (7 hours a week) in Ring III drop mathematics, drawing, and one hour of English; in Ring IV Greek (7 hours) is then substituted for mathematics and drawing.

As to mathematics, instruction in a realgymnasium includes: (a) Algebra-theory of indices, logarithms, arithmetical and geometrical series, compound interest; (b) geometry-proportion applied to geometry, problems (especially in plane mensuration), solid geometry; (c) plane trigonometry-simple computations in connection with right and oblique triangles; (d) analytic geometry-curves of the second degree; the notion of a derivative is made clear and much emphasis is laid on graphic representation of functions.

The course in a gymnasium is concluded with a final examination (studentexamen, or afgångsexamen, or maturitetsexamen) which, in either "side," entitles those who have passed it to matriculation at the universities. Examination commissioners called "censors" are appointed by the Government to superintend each examination.

They are chosen for the most part from professors of the universities, and usually number about 18. The censors are assisted by "examiners," who are the teachers of the schools; the examiners always act under the direction of a censor and in his presence. Questions for the written part of the examination are drafted by the censors but must be approved by the royal board of secondary schools. This examination takes place simultaneously in all the schools some weeks before the oral examinations and lasts four or five days.

The written examinations in both "sides" include Swedish, modern language, and mathematics, and in the realgymnasium physics, in the Latingymnasium Latin. French may be substituted for either of the "modern languages," which are German and English. The single examination in mathematics lasts six and one-half hours. The paper contains eight or nine questions. An adequate discussion of three at least is necessary for passing.1 The answers are first looked over by the teacher himself and one colleague, and are graded with one of six predicates, the fifth of which in descending order is "satisfactory," and the sixth "unsatisfactory."

The candidate is not admitted to the oral part of the examination unless in each of his papers he gains at least the mark "satisfactory." When the teachers have thus arrived at their decisions, the papers are stitched together into books and sent to the royal board, which distributes them among the censors for their inspection, much of which they make as they travel in the train from school to school, from one end of Sweden to the other. The final decision is not reached until censors and teachers sit together in conclave somewhat later.

The oral examination is wider in range than the written examination and embraces all the subjects taught in the later years of school life. The rektor or headmaster in each school arranges the candidates in groups of five or six in a room; and in these rooms they are kept throughout the examination, while the censor is sometimes in one room and sometimes in another, according to the subjects of the examination: Various arrangements emphasize the importance of the occasion. At least three persons of position, nominated by the inspecting local authority, represent the public. Evening dress is de rigueur for censors, examiners, and candidates; and the censors and the teachers wear their robes and decorations. The teacher of the candidates is generally the examiner; and, as a rule, puts all the questions under the direction and guidance of the censor. The examination, though apparently of the pupil alone, is really quite as much a test of the teacher, enabling the censor to judge of the efficiency of his work. It is also deemed useful for university professors to be brought for a few days in each year into such close touch

1I quote freely from Mr. Thornton in what follows.