XIII. THE WORK IN MATHEMATICS IN THE TWELFTH SCHOOL YEAR.

AUSTRIA.-The twelfth school year is the eighth and last year of the Gymnasium and of the Realgymnasium.

Gymnasium.-The mathematics of previous years is thoroughly reviewed, and various ideas are studied more intensively. Practical applications are emphasized, and a good deal of attention is devoted to the history of the subject and to the introduction of general ideas of means in advanced mathematics. In both the Gymnasium and Realgymnasium the elements of differential and integral calculus are presented. The subject matter is treated in close relation with the theory of functions as presented in previous years. Special attention is given to the following functions: ax+b, ax2+bx+c,

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a

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log x

ax3, √x, sin x, cos x, az and Problems are given to determine ax+b' x2' the tangents of curves, and velocities; maxima and minima are also studied. BELGIUM.--The twelfth school year is the sixth school year of the Athénée Royal. In the Greek-Latin and in the commercial courses three hours a week are devoted to mathematics. In the Latin and the scientific courses six hours are devoted to mathematics.

Greek-Latin course.-The square and square root, radicals, quadratic equations, equations reducible to quadratics, special quadratics of higher degree, arithmetical and geometric progression, proportion, logarithms, interest, and annuities are studied. The geometry of the previous year is reviewed, and solid geometry is completed. Trigonometry is introduced. Tables are used, and the formulas for the right triangle are developed and applied in numerous problems.

Scientific and Latin courses.—The algebra and plane geometry of previous years are reviewed, and trigonometry and its applications are studied. The different systems of numeration are considered, and computations involving various number scales are made. Short methods for multiplication and division are emphasized.

The application of indeterminant coefficients in functions, and the relation between the coefficients of algebraic equations in order to satisfy certain given conditions are studied. Certain problems in maxima and minima are given. Continued fractions are presented. Indeterminate equations of the first degree, permutations and combinations, the binomial theorem, the summation of series, powers and roots, radical equations, exponential equations, and Naperian logarithms are studied.

The course in plane geometry includes the study of functions, harmonic pencils, poles and polars, and the theorems of Pascal and Brianchon concerning the circle. The course in solid geometry includes the general subject matter of Books VI, VII, VIII, and IX.

DENMARK.-The twelfth school year is the third and last year of the Gymnasium. In the classical and in the modern language courses two hours a week are devoted to the study of mathematics, and in the mathematics and scientific courses six hours are devoted to the subject.

The course includes a comprehensive review of the mathematics of previous years. The elements of analytic geometry are introduced. The pupils are taught how to determine points and curves by means of rectilinear and polar coordinates. The formulas for the equation of the straight line and circle, tangents, parabola, ellipse, and hyperbola are developed; and the principal theorems for tangents, normals, and asymptotes are studied.

In addition to this, the pupils elect one of the following courses:

(4) Determinants with their applications to linear equations, continued fractions and their applications, the general equation of the second degree treated analytically, the icosahedron and dodecahedron and the representation of similar polyhedra by

means of vertical projections on perpendicular planes, and the plane sections of these solids.

(B) Infinitesimal calculus. The course includes the computation with infinitesimal magnitudes as an introduction to differential and integral calculus, the continuity of fractions, the derived function of xn, (n being rational) and of the trigonometric functions, of the sum, product, and quotient, and of functions of a function, the theorem of Rolle, and maxima and minima. Taylor's theorem for integral functions is presented. Definite and indefinite integrals are considered, and the simplest functions are integrated. Simple applications are made to geometry and physics.

ENGLAND.—The instruction in the central schools usually closes with the eleventh year. In the great private schools, such as Eton, Harrow, Rugby, and Winchester, there are boys from 13 to 19 years of age. Many of these boys go to the universities or enter the army. In most of the great private schools, there are three courses:

(a) Classical course, which corresponds roughly to the German Gymnasium. (b) The modern course, which corresponds roughly to the German Realgymnasium.

(c) The army course.

Courses (a) and (b) are usually about equal in number of students. Course (c) usually has a smaller enrollment.

In the modern course from four to six periods a week are devoted to the study of mathematics. Details of this course are not available in the English reports. Great emphasis is put upon the entrance requirements in mathematics by Oxford and Cambridge, and more time is devoted to the subject in the classical courses in England than in corresponding courses in France or Germany.

FRANCE. The twelfth school year is usually called Classe de Philosophie.

The relations between algebra and geometry are emphasized, and a good deal of attention is devoted to the subject of graphs. The pupils are taught how to construct a rectangle which has a given side and is equivalent to a given square, and numerous other similar constructions. The notion of coordinates is extended, and the study of functions is continued. Squared paper is used in the determination of areas. The theory of limits is discussed. The formula for the area of a parabola is derived. The application of the methods of infinitesimal calculus to the evaluation of surfaces and volumes of figures considered in elementary geometry is made.

GERMANY. The twelfth school year is the ninth and last year of the Bürgerschule, Gymnasium, Realgymnasium, Realschule, and Oberrealschule.

Gymnasium.—The formulas of elementary trigonometry are applied to mathematical geography and to elementary astronomy. A comprehensive review of the mathematics of previous years is given, and applications to physics are especially emphasized. The concept of coordinates is introduced and applied to the study of conics. Realgymnasium and Realschule.-Five hours a week are devoted to the subject of mathematics and two to physics. The subjects are very closely correlated. Three hours are usually devoted to the study of plane analytic geometry and a comprehensive review of the mathematics of previous school years. The study of analytics includes the point, straight line, and circle. Two hours are usually devoted to the study of descriptive geometry. Pyramids, prisms, cones, cylinders, and spheres, and sections of these bodies, are presented. The ellipse and parabola are usually not considered. In some schools cubic equations and maxima and minima are studied. Oberrealschule.-Five hours a week are devoted to the study of mathematics. The course includes both analytic geometry and infinitesimal calculus. Logarithms and exponential functions with their derivatives are studied. Indefinite and definite integrals with simple exercises are introduced. The principles of calculus are applied to exercises in mechanics.

The course in geometry includes the study of conics treated both analytically and synthetically. These curves are regarded as sections of right circular cones.

The theorem of Quetlet-Dandelin is introduced. The relation between algebra and geometry is especially emphasized.

The course in descriptive geometry is a continuation and extension of the work of the eleventh school year.

A comprehensive review of the mathematics of previous years is given, and the historical development of the subject is especially considered.

HOLLAND.-The twelfth school year is the sixth year of the Gymnasium. It is sometimes called the preparatory year. Plane and spherical trigonometry are studied, and the elements of analytic geometry are introduced. A comprehensive review of the mathematics of previous school years is given. Especial emphasis is put upon the study of permutations and combinations, the theory of probability, determinants, continued fractions, higher series, and functions. The correlation between mathematics and physics is emphasized.

HUNGARY.--The twelfth school year is the eighth and last year of the Bürgerschule, Gymnasium, and Realgymnasium.

In the Gymnasium and Bürgerschule two hours a week are devoted to mathematics, and in the Realschule three hours are devoted to the subject of mathematics and two to descriptive geometry.

Gymnasium and Bürgerschule.-Permutations and combinations, the theory of probability, the binominal theorem, and Pascal's triangle are studied, and a comprehensive review of algebra is made. The formulas of spherical trigonometry are applied in the study of geography. The elementary ideas of coordinates are introduced. The study of free-hand drawing is continued and a thorough review of geometry is made. Realschule.-A comprehensive review of the algebra of previous years is made. The idea of coordinates is introduced, and the principal equations of points, distances, straight lines, and circles are developed and applied. The ellipse, parabola, and hyperbola are studied as geometric loci, and the entire subject of geometry is thoroughly reviewed.

The course in descriptive geometry includes the study of orthogonal projections, the representation of cones, cylinders, spheres, and the intersection of these bodies by lines and planes, tangent planes, shadow constructions, and a thorough review of the work of previous years.

ITALY.—The twelfth school year is the third and last year of the liceo. No mathematics is offered during this year.

JAPAN. In most of the schools the middle school closes with the eleventh school year. When an additional year is offered, the time is usually devoted to a comprehensive review of the mathematics of previous years, and to a more intensive study of the subject.

ROUMANIA.-The twelfth school year is the fourth and last year of the lycée. Five hours a week are devoted to the study of mathematics.

The course in algebra includes the theory of roots, theorem of Rolle, Descartes's theorem, the solution of equations of higher degree, the study of integral, fractional and irrational roots, the methods of approximation, and other elementary ideas of the theory of equations.

The course in analytic geometry includes the study of rectangular coordinates, and problems concerning the straight line, circle, ellipse, hyperbola, and parabola, and their properties. A good deal of emphasis is put upon the study of cosmography, and mathematics is closely correlated with this subject.

RUSSIA. The course in Russia closes with the eleventh school year.

SWEDEN.-The twelfth school year is the fourth and last year of the Gymnasium. In the Latin Gymnasium five hours a week and in the Realgymnasium six hours are devoted to the study of mathematics.

Latin Gymnasium.-The course includes a comprehensive review of the mathematics of previous years and the introduction of the elementary ideas of analytic geometry. Linear drawing is taught, and the principles of perspective are emphasized.

Realgymnasium.-The concept of derivatives is introduced and applied to finding maximum and minimum points and to the determination of equations for perpendiculars. Only simple expressions, such as y=x2, y=x3, y=√x, y=3√x, and y=sine x are developed.

SWITZERLAND.-The twelfth school year is the sixth year of the Gymnasium and the fourth of the Realschule.

In the classical Gymnasium four hours a week, and in the Realgymnasium four and a half hours, are usually devoted to the study of mathematics. In the Realschule nine hours a week are frequently given to the subject.

In the Realgymnasium two hours are usually devoted to algebra, one and a half to geometry, and one to descriptive geometry. In the Realschule three hours are devoted to algebra, two and a half hours to geometry, and three and a half hours to descriptive geometry.

Gymnasium.-The course includes the solution of difficult quadratics and the study of permutations and combinations, the theory of probability, and the binomial theorem. The correlation of mathematics with mechanics and physics is emphasized. In some Gymnasia the elementary ideas of analytic geometry and of infinitesimal calculus are introduced.

Realschule. The course includes the study of series, compound interest and annuities, permutations and combinations, and the binomial theorem. The study of analytic geometry is introduced and the fundamental formulas for a point, straight line, ellipse, parabola, and hyperbola are developed and applied. In most of the Realschulen the elements of infinitesimal calculus are introduced. In some of the Realgymnasia and Realschulen de Moivre's theorem and operations with complex numbers are introduced. Approximate methods for the solution of numerical and indeterminate equations, continued fractions, maxima and minima, and spherical trigonometry are also taught in a few of the Realgymnasia and Realschulen. In many of the Realschulen the elements of solid analytic geometry are also taught. About one-third of the Realschulen take up the general equation of the second degree with two variables. The study of conic sections is quite thorough in most of the Realschulen. In the courses in differential and integral calculus the transition from differences to derivatives is carried out by objective geometric methods in every case. Exponential and logarithmic functions are developed, and the rules for the derivative of the sum, product, and quotient, and of a function of a function are taught. The equations of tangents of the conic sections and of other curves are studied. In a few schools differential, but not integral, calculus is taught. In some schools differential calculus is taught, but only the merest elements of integral calculus are studied. In most of the schools the elements of infinitesimal calculus have been taught for many years.

The course in descriptive geometry includes the construction of plane sections, of the regular solids and their intersections; also the representation of right cylinders and cones and tangent planes.

THE UNITED STATES.-The twelfth school year is the last year of the secondary school. The courses in mathematics vary somewhat, but in most of the schools plane trigonometry is studied during the first half of the year and college algebra during the second half. In many of the schools the application of trigonometry to surveying is emphasized, and some practice in actual surveying is given.

Spherical trigonometry is usually not included, but it is not unusual for the course to include the study of the right spherical triangle.

The course in college algebra usually includes a thorough review of quadratic equations with two or more unknowns, graphs, proportion, the progressions, the binomial theorem for positive integral exponents, determinants, inequalities, permutations and combinations, the theory of probability, and methods for the approximation of roots. The mathematics of the twelfth school year is always elective.