The materials of the program, which are about the same for all universities, have been arranged in the following manner by Mr. Rose: (a) PURE MATHEMATICS. 1. Analysis—First year.-Differential and integral calculus. Three hours a week. Limits; aggregates; derivatives and differentials; Taylor's and Maclaurin's theorems; explicit and implicit functions; change of variable; maximum and minimum; series; geometric applications of differential calculus to curves and to surfaces; integrals-processes of integration; various types of integrals; areas, surfaces, volumes. Second year.-Definite integrals, differentiation and integration; Eulerian integrals; differential equations; integrable types; simultaneous differential equations; partial differential equations of the first order; total difference equations; calculus of differences and calculus of variations. Three hours a week. Third year.-Theory of a complex variable; synectic functions; study of works of Abel, Cauchy, Riemann, Weierstrass, and their disciples. Theory of elliptic functions (after Legendre). Three hours a week. Fourth year.-Six hours a week, and more for students working for their doctorate in analysis. Searching study of a topic in the theory of functions. Elliptic functions according to Jacobi and Weierstrass. Research in differential geometry based upon the work of Darboux' and Bianchi. 2 The masterly work of M. de la Vallée-Poussin' gives a good idea of the subjects covered in the first two years. 2. Analytic Geometry-First year. Three hours a week. Revision of analytic geometry of two dimensions and study of analytic geometry of three dimensions. Particular study of homogeneous, tangential, triangular, and tetrahedral coordinates. Generation of surfaces. Surfaces of the second degree. For such work the notable treatise by Servais of the University of Ghent may be consulted. Second year. Three hours a week. Projective geometry: Study of forms, involution, homography, homology, correlation, duality, polarity, properties and generation of conics, pencils, nets, generation of quadrics, properties. The texts of F. Folie, F. H. G. Deruyts, and of Servais illustrate the requirements. 4 For the pupils who specialize in geometry during their third and fourth years, the professor takes up either the theory of plane and cubic curves and of cubic surfaces, or the theory of forms in higher geometry. The number of hours per week depends on the professor. The works of F. Folie, F. H. G. Deruyts, M. Stuyvaert, Fairon, and L. Godeaux may be mentioned. 1 G. Darboux: I. Leçons sur la théorie générale des surfaces. 4 parties. Paris, Gauthier Villars. 1:2e éd., 1914,7+618 pp. 2:2e éd., 1915, 3+579 pp. 3:1894, 8+512 pp. 4:1896, 8+548 pp. II. Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. 20 éd. Paris, Gauthier Villars, 1910. 3+567 pp. * L. Bianchi, Lezioni di geometria differenziale. 3 vols. Pisa, Spoerri, 1902-1909 (2d edition of volumes 1-2). Second German edition by M. Lukat. Leipzig, Teubner, 1910. 18+721 pp. C. J. de la Vallée-Poussin, Cours d'analyse infinitesimale. 2 tomes. Louvain, Dieudonné, 1:30 éd. 1914; 2:2e éd. 1912. 9+452+9+464 pp. F. J. P. Folie, Eléments d'une théorie des faisceaux. Bruxelles, Hayez, 1878. 110 pp. Géométrie supérieure cartesienne, Bruxelles, Hayez, 1872. C. Servais: 1. Sur les imaginaires en géometrie, application aux cubiques gauches. Gand, 1894. 2. Cours de géométrie analytique. 2e éd. Gand, 1906. J. Fairon, Sur les involutions du quatrième ordre. Bruxelles, Hayez, 1909, 68 pp.-M. Stuyvaert, "Recherches relatives aux connexes de l'espace" (1901) and "Étude de quelques surfaces algébriques engendrées par les courbes du deuxième ou du troisième ordre" (1902).-F. Deruyts, "Mémoire sur la théorie de l'involution et de homographie unicursale" (1890). (a) PURE MATHEMATICS-Continued. 3. Algebra-Thorough study of determinants. General theory of equations; resolution and methods of approximation to the roots, study of imaginaries. Two hours a week in the first year. Texts: Algebra by J. Neuberg or by P. Mansion. 4. Calculus of Probabilities—Third year. One hour a week. Principles and problems; various species of probabilities. Bernoulli's theorem; theory of play; law of large numbers; theory of least squares; application to annuities and life insurance. Text by Boudin.' (b) APPLIED MATHEMATICS. 5. Astronomy-Second year. Physical astronomy. Three hours a week. Study of the system of the world, systems of coordinates, motions, sun, moon, planets, stars, comets. Applications of spherical trigonometry, elements of geodesy. Text: Astronomy (Collection Léauté) by Stroobant.2 Third year. Three hours a week. Mathematical astronomy, application of analysis to astronomy, refraction, eclipses, calculation of orbits. In the fourth year the students make a thorough study of some branch of mathematical astronomy. 6. Descriptive Geometry-First year. Four hours a week. Review of the principles of the point, the line, and the plane. Study of trihedral angles, of curves and of surfaces, surfaces of the second degree and ruled surfaces, intersections, géométrie cotée. Treatises: By Chomé,3 Breithof, de Locht, Van Rysselberghe, and Chargois. 7. Rational Mechanics-First year. One hour a week. Vector geometry and statics: Forces, equilibrium, virtual velocities, funicular curves, machines. Second year. One hour a week. Kinematics: Velocity and acceleration, finite motion, instantaneous motion, continuous motion. Third year. Dynamics. Three hours a week. Study of the motion of free point, of a point on a surface or on a curve. Relative motion. D'Alembert's principle and the general equations of dynamics (Lagrange and Hamilton). Motions of a system. Rigid systems. Percussions. Attraction of ellipsoids. Mechanics of fluids; hydrodynamics. Text: The remarkable work by Massau." Students who continue the study of mechanics during the fourth year take up equations of mechanics and the principal theories of celestial mechanics. 8. Mathematical Physics—Third year. Three hours a week. Study of the principal theories of optics, of magnetism, and of electricity. (c) 9. HISTORY OF MATHEMATICS. Third year. Two hours a week. Mathematics in antiquity among the Egyptians, Assyrians, Chaldeans, Greeks, and Romans. Mathematics of the Hindus and Arabs. Middle ages. Fourth year. Two hours a week. Renaissance, modern times, contemporary history, detailed study of each of the branches: Arithmetic, algebra, geometry, analysis, mechanics, physics. 1 E. J. Boudin, Leçons sur le calcul des probabilités. The first edition was published anonymously in autograph form in 1865; 3e éd. (same as second). Gand, De Witte, 1889. Autographie, 125 pp. *P. Stroobant, Précis d'astronomie prátique. Paris, Masson, 1903. 188+16 pp. * F. Chomé: 1. Cours de géométrie descriptive. 3e édition entièrement revue, corrigée et augmentée. 3 vols. Bruxelles, 1898-1904. 2. Éléments de géométrie descriptive. Bruxelles, Kouwenaar, 1896. 12+159 pp. N. Breithof, Cours de géométrie descriptive: surfaces, courbes. 2 vols. Louvain, 1875. J. Massau, Cours de mécanique. 2 vols. Gand, Meyer Van Loo, 1892-1896. (Autographié.) 365+ 8+312 pp. Leçons de mécanique rationnelle, 2 tomes. Gand, L'association des ingénieurs sortis des écoles spéciales, 1911-1913. 15+259+17+343 pp. (c) HISTORY OF MATHEMATICS-Continued. 10. Elementary Mathematics. In the course on methodology (three hours a week, fourth year) the principles and foundations of such matters are considered. Review of the principal theories studied in the athénée with a view to practical lessons. Notions of higher arithmetic, of various kinds of geometry, of transcendental numbers. Text: Methodology, by Dauge.1 (d) OTHER COURSES. 11. General Physics-First year. Four hours a week. Study of the principal 13. Psychology, Logic, and Moral Philosophy-Second year. Three hours a week. 14. Crystallography-Second year. Three hours a week. 2 Systems, properties, Students in a university have to pass annual examinations on each of the subjects of study during the year before being admitted to the work of the following year. Having satisfactorily completed the first two years of work they receive diplomas as candidats. The tests at the end of the fourth year include: (1) The presentation and public defense of a thesis; and, for those who are to become teachers, (2) the public delivery of two lessons, one on mathematics, the other on experimental physics. The subjects of these lessons are given in advance by the jury and are chosen from the program of the athénées. All tests having been successfully passed, the candidate becomes a doctor of physical sciences and mathematics. The examinations occur each year in July and in October, and there are several grades of diplomas: With success, with distinction, with great distinction, and with greatest distinction. PROFESSIONAL PREPARATION. It is noteworthy that the program for the doctorate includes the elements of the history of mathematics and a course on methodology of the teaching of mathematics and of physics. This latter course deals equally with subjects taught in the athénée and with the methods of mathematics in general. The course lasts one or two years (third and fourth) and averages about three hours a week. The lessons are conducted by a university professor who has generally been a teacher in the secondary schools in his earlier career. They have a bearing on the methods of teaching each of the parts of the program of the athénée, and the professor usually expounds each of these subjects through the medium of the students themselves, aided by his counsel and advice. Each student gives before his fel 1 F. Dauge, Cours de méthodologie mathématique. 2e [last] édition. Gand, Hoste, 1896. 10+525 pp. The pass mark is 50 per cent. lows practical lessons in mathematics and physics. These are afterwards passed under critical review by professor and students. While students are required to take a course in psychology, pedagogy is not taught at all systematically. It is with such scientific and professional preparation that the future teacher in the better secondary schools enters upon his duties. TEACHERS IN SECONDARY SCHOOLS. In the athénées the teaching staff consists of an inspector of studies (préfet des études), professors, and masters (surveillants). The head of a lower middle school is called a rector. The inspectors, rectors, and professors are nominated by the King, and must each have secured the doctor's degree at a university. The masters, who are chosen from candidates, are appointed by the minister of sciences and arts. In general the newly made doctor enters first either (1) as professor in a free school (établissement libre) or communal college; or (2) as temporary or permanent surveillant; or (3) as substitute professor in an athénée. After some years have passed in one or another of these capacities, he may be promoted to a chair in an athénée; but in many cases the doctor must proceed to this goal by way of the position as surveillant. The mathematical chairs vary in attractiveness, according to the divisions: (A) Greek-Latin, (B) Latin, (C)-(D) modern humanities, with which they are connected. In establishments of secondary importance (averaging about 200 pupils) there are ordinarily three professors of mathematics, one for division (A) in VII and VI, the course being the same for the divisions (A) and (B); a second for the modern humanities VII, VI, V, and IV, and for (B) V and IV; finally a third for division (C): III, II, I scientific, and division (B): III, II, I. There is only one corresponding professor in each athénée; he is the professeur de mathématiques supérieures. So, also, there is always only one professor in division (A). On the other hand, the number of professors in the division of modern humanities may be two or three and sometimes four, according to the number of pupils (400 to 700). But in any case as there are only 20 athénées, and a smaller number of similar ranking collèges communaux, the number of mathematica chairs is relatively limited. The number of teaching hours per week varies from 18 to 21, according to the divisions. The salary is composed of two parts: (1) A variable part, from the minerval, which accrues from equal distribution among the professors of fees paid by the pupils; and (2) a fixed part. 2 This term is applied to the fee paid by the pupils for scholastic instruction. If the minerva' part does not amount to at least 700 francs a year, the State makes up the deficiency. In larger athénées this part of the salary may range from 900 francs to 2,000 francs, or even more. As to the fixed part of the salary, the initial amount is 2,600 francs. By periodic increments it may reach 5,500 francs in the following manner: Francs. Initial salary, 2,600 Francs. Surveillants commence with a salary of 2,200 francs, but have an increase of 200 francs every three years; the years passed as surveillant or as substitute teacher count in fixing the salary of the teacher, who finally becomes a professor. In the collèges communaux the initial salary varies from 1,800 to 2,400 francs; the increases vary according to the schools, and the minerval is not distributed among the professors. The years spent in a collège communal are always counted toward promotion when a professor is called to an athénée. At the head of each athénée is a préfet des études who does not teach and who has been chosen from among the professors, at least 40 years of age, in another establishment. Apart from the variable minerval the salary of a préfet ranges from a minimum of 4,400 francs to a maximum of 5,900 francs; he has also free residence, heat, and light. The chairs at athénées of large cities are most sought after, because of the higher minerval and the attractions which large centers offer. As a general thing professors of mathematics start in division (A) or in division (D), and after some years pass to division (C) if they have acquired distinction by their professional aptitude and their publications. There is no definite rule concerning advancement, though the rule of seniority is ordinarily respected. Every professor 60 years of age is retired with a pension. This pension may be obtained when he is 55 years old if he has taught for 30 years, or if he has had to give up work on account of disability. The basis of calculating the pension is the average salary, minerval included, for the last five years of service. The pension is 1/55 of this amount for every year of service, including the four years of study. Thus a professor beginning with any title in secondary-school work at the age of 24, and pensioned at 60 years, counts first 36 years of service, then the 4 years at the university. He has then the right to 40/55 of his average salary, say (40/55) (5500+700) = (40/55) × |