secondary education, the whole field of elementary algebra, plane and spherical trigonometry with analysis of circular functions. The second course deals with higher algebra; and the general theory of equations, as well as an introduction to modern higher algebra, is presented. During one year at the University of Madrid, for example, this introduction dealt with the theory of binary forms (discriminants; Jacobians, Hessians, and Wronskians; linear substitutions; invariants and covariants; canonical forms).

2. Metric geometry.-This geometry is simply the ordinary Euclidean geometry modified by the use of modern synthetic methods. The course in the faculty of sciences is divided into two parts: The first deals with fundamental theories, those on which the theorems of all previous questions are based; the second consists in studies of each group of figures in detail, without losing sight of the groupings which facilitate the employment of the principles of duality, projectivity, etc.

(a) Fundamental theories: First notions; angles; perpendiculars and parallels; relation between the elements of triangles and of trihedral angles, isosceles or scalene; distances and inclinations large and small; elementary cases of symmetry and of equality of triangles, trihedrals, and tetrahedrons; equality and orthogonal symmetry in general; geometric loci; circle, cone, and sphere; proportional lines; similar figures; products of segments in triangles; concept of length of curves in general and of the areas of conics; plane and spherical trigonometry sufficient for the solution of right triangles; orthogonal projections.

(b) Studies: More complete consideration of each of the following figures: Ranges of points, pencils of rays or of planes; their cross ratios and projection; segments and angles; triangles and trihedral angles; quadrilaterals, tetrahedral angles, and tetrahedrons; ordinary polygons, polyhedral angles, and polyhedrons in general, and in particular those which are regular and semiregular; prisms and cylinders; pyramids, prismatoids, cones, and spherical figures; systems of circles, of cones, and of spheres; particular determination of length, of areas, and of volumes; their comparison and proportionality; homology, homothety, involution, and symmetry in general; polarity of a circle, of a cone, or of a sphere; inverse figures; stereographic projection. 3. Analytic geometry.-This course is based on projective methods and is especially developed at Madrid.1

4. Elements of infinitesimal calculus.-This course includes a discussion of theory of limits, continuity, orders of infinitesimals derivatives and differentials of functions of one or of several variables; change of variables; hyperbolic functions; Legendre's polynomials and developments into series by formulæ of Taylor, Maclaurin and Lagrange; maxima and minima; elements of integral calculus; differentiation and integration under the integral sign with application to the calculation of definite integrals and of the Eulerian integrals. In some programs this first part of the calculus course concludes with a discussion of curvilinear and surface integrals and of integrals in the formula of Green, of Stokes, and of Dirichlet, and their applications in mechanics and physics.

5. Cosmography and physical geography.

6. Geometry of position.

7. Descriptive geometry. 8. Rational mechanics.

9. Spherical astronomy and geodesy.

The title of licenciado in mathematical science is obtained by a candidate after three examinations are successfully passed: (a) A written examination on two questions drawn, by lot, from those proposed by the tribunal; (b) an oral interrogation by the three

J. REY PASTOR, Fundamentos de la geometria proyectiva superior. Tomo 1, Madrid, 1916. 22+444 pp. Cy. Bulletin des sciences mathématiques tome 41, 1917 p. 229.

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. 26me 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:

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.