to quadratics, symmetrical equations, simultaneous quadratics, powers and roots, and annuities are also studied. The subject of geometry is elective, and the course is practically the same as in the corresponding year of the Gymnasium.

Gymnasium.-In the classical Gymnasium the course includes the study of general quadratic equations, the rules for the sum and the product of the roots of quadratic equations, inequalities of the second degree, maxima and minima, limits, simultaneous quadratics, imaginaries, four-place logarithms, simple exponential equations, arithmetical and geometrical progression, infinite series, compound interest, and annuities.

The course in geometry includes the study of similar figures, regular polygons, the study of the circle, the trigonometric functions of acute and obtuse angles with simple applications, and the graphic representation of simple functions.

The course in the scientific Gymnasium includes all that is offered in the classical Gymnasium, with the addition of the study of approximate computations, harmonic progressions, complex numbers, permutations and combinations, indeterminates with two unknowns, and complex roots.

ENGLAND. The tenth school year is the last year of the five-year preparatory course. The work in arithmetic includes the mensuration of plane and solid figures and the study of simple business arithmetic.

The course in geometry includes simple constructions, loci problems, and propositions concerning parallels, the congruency and similarity of triangles, areas, and proportional parts.

The course in algebra includes the theory of exponents, logarithms, and variations. The course in trigonometry includes the numerical solution of triangles by means of the sine and cosine formulas, the use of tables, and simple problems from surveying. The course in girls' schools for the corresponding year is more elementary. The course in arithmetic includes the study of general arithmetic, compound interest, stocks and bonds. The course in algebra includes quadratic equations, the theory of indices, surds, progression and proportion. In geometry, Books I to IV of Euclid are usually completed.

FINLAND.-The report makes no division of the work by years. Algebra is begun in the third year of the lycée, geometry in the fourth, and trigonometry in the fifth. The subject of solid geometry is somewhat abridged. The tenth school year is the sixth year of the lycée.

FRANCE. The tenth school year is the sixth year of the lycée. It is the second and last year of the second cycle. In the classical lycée two hours a week, and in the scientific lycée five hours are devoted to the study of mathematics.

Classical lycée. The course in algebra includes exercises involving equations of the first degree and of the second degree in one unknown, variations of trinomials of the second degree.

The course in geometry includes the mensuration of angles, the study of plane figures, the definition of sine, cosine, and tangent for angles from zero to 180 degrees, and the formulas for surfaces and volumes.

Scientific lycée.-The course in algebra includes the study of equations of the second degree, applications of geometry to trigonometry, equations of derivatives of simple functions, the study of variations and of graphic representations, the study of motion in connection with the theory of derivatives.

The course in geometry includes the study of polyhedra, cones, spheres, surfaces, and volumes. The ideas of translation and rotation are extensively used and the subject of symmetry is given a good deal of emphasis. Rather an extensive course in descriptive geometry is offered in many schools. Geometric drawing is also taught in most of the scientific schools. The trigonometric functions are studied, and the principal formulas for the solution of right and oblique triangles are derived.

GERMANY.-The tenth school year is the seventh year of the Bürgerschule, the Gymnasium, Realgymnasium, Realschule, and Oberrealschule.

Gymnasium.-Equations of the second degree, including simultaneous quadratics and equations in the quadratic form, are studied. Four and five place logarithmic tables are also used. Emphasis is put upon the application of algebra in the solution of geometric problems.

Geometry. The course in geometry includes the study of congruent and similar figures, inscribed and circumscribed polygons, cyclometry, and harmonic functions. The plane trigonometry of the right and oblique triangle and goniometry are introduced. In some schools harmonic points and rays are studied.

Realgymnasium.-Six hours are devoted to mathematics, three to algebra, and three to geometry. Arithmetical and geometrical progression, compound interest and annuities, are studied. Equations of higher degree, difficult quadratics, reciprocal and binomial equations are introduced.

The course in geometry includes the study of straight lines, planes, triangles, polygons, and circles. The principal properties of the pyramid, cylinder, prism, cone, sphere, and logarithms are studied, and the study of trigonometry is continued. In a few Realgymnasia the elements of analytic geometry are begun. The straight line, poles and polars, the ellipse, parabola, and hyperbola are considered.

Realschule and Oberrealschule.-Two hours a week are devoted to the subject of algebra, and three to geometry and trigonometry.

The course in algebra includes the study of imaginary and complex numbers, De Moivre's theorem and its application, the graphic solution of equations, the binomial theorem, and reciprocal equations.

The course in geometry includes the study of similar figures; the functions for any angle are taught and are represented by line values; and the applications of spherical trigonometry to solid geometry and to mathematical geography are presented. In some schools map projection is studied. In most of the schools a brief course in descriptive geometry is offered. This includes a systematic review of the descriptive geometry of previous years, and in addition the study of the intersection of planes and solids by other planes and solids.

HOLLAND.—The tenth school year is the fourth year of the Gymnasium and of the middle school.

Middle school.-Four hours a week are devoted to the study of mathematics. The algebra of previous years is reviewed. Indeterminate equations of the first degree are introduced. Permutations and combinations and the binomial theorem are studied. Trigonometry is introduced. The work in solid geometry includes a study of the relative position of straight lines and planes, dihedral and polyhedral angles, the properties and volumes of prisms, pyramids, cylinders, cones, and spheres. A course in the elements of descriptive geometry is usually offered. Two hours a week are usually devoted to the study of mechanics and one to cosmography.

Gymnasium.-Three hours a week are devoted to the study of mathematics, one hour being given to algebra and two to geometry. Irrational and complex expressions, fractional and negative exponents are studied. All of plane geometry is completed.

HUNGARY.-The tenth school year is the sixth year of the Gymnasium and of the Realschule.

Gymnasium.-Involution, involving fractional and negative exponents, is taught. Brigg's logarithms are used, and arithmetical and geometrical progression are studied. Most of the course in geometry is devoted to the study of the circle.

The functions of acute angles are taught and used in the solution of triangles. Formulas for the sine and cosine are developed and applied. Functions of positive and negative angles are considered. Some simple surveying is done.

Realschule.The courses in algebra and in geometry are the same as in the Gymnasium, but the work is somewhat more intensive.

Three hours a week are devoted to the study of descriptive geometry. The course includes the study of orthogonal projections, the projections of circles in simple positions, the study of points, straight lines and planes, angles, and shadow problems.

Bürgerschule.—Three hours a week are devoted to the study of mathematics. Linear equations, square and cube root, quadratic equations, and involutions with fractional and negative exponents are taught. Some attention is given to irrational and imaginary numbers. Logarithms, the theory of combinations, and annuities are also presented.

The course in geometry includes the study of the Pythagorean theorem and its applications, and the study of the circle. In some classes one hour a week is devoted to simple bookkeeping.

ITALY.-The tenth school year is the first year of the liceo. Six hours are devoted to the subject of mathematics, considerable attention being devoted to arithmetic. The study of geometry includes a review of the first book of Euclid, which was studied in the preceding year. Books II and III are also completed. The study of algebra is completed up to radicals and fractional exponents. The study of arithmetic is completed with the theory of square root and incommensurable numbers.

JAPAN. The tenth school year is the fourth year of the middle school. Two hours a week are devoted to algebra and two to geometry. The course in algebra includes the study of irrational expressions, ratio and proportion, the progressions, permutations and combinations, logarithms, the use of tables, and the binomial theorem with positive integral exponents.

The course in geometry includes the study of inscribed and circumscribed figures, areas, congruency, proportion, and the applications of proportion to similar figures. ROUMANIA. The tenth school year is the second year of the lycée.

Algebra. The functions of independent variables, limits, continuity, simple derivatives, graphs, maxima and minima, the theory of permutations and combinations, and the binomial theorem are studied. In some schools, the extraction of the square root of polynomials by indeterminate coefficients is taught. In a few schools, determinants of the third order, and their use in the solution of homogeneous and nonhomogeneous equations are taken up. Graphs are extensively used in all courses in algebra.

Geometry. The course in geometry includes the study of regular polygons, circles, transversals, areas, congruency, and similarity. Some attention is devoted to the study of harmonic functions. Many applications are presented. Dihedral and trihedral angles, parallel and perpendicular planes, and the formulas for surfaces and volumes are studied.

Trigonometry. The trigonometric functions for acute and obtuse angles are taught and represented graphically. The principal formulas of plane trigonometry are developed and applied, and the tables of logarithms are used. The idea of periodic functions is emphasized.

RUSSIA.-The tenth school year is the seventh year of the Realschule and of the Gymnasium.

Gymnasium.-Three to four hours a week are devoted to the study of mathematics in the Gymnasium. The course in algebra includes simple and quadratic equations involving one or more unknown quantities, indeterminant equations of the first and second degrees, the progressions, logarithms, the binomial theorem, and continued fractions.

Trigonometry. One and a half hours to two hours are devoted each week to the study of trigonometry. Plane trigonometry is completed during this year.

Realschule. The course in algebra in the Realschule is practically the same as in the Gymnasium, but the work in the Gymnasium is somewhat less intensive. In the

Realschule more emphasis is put upon the subject of complex numbers, and upon integral functions and their roots. The course in trigonometry is the same as in the Gymnasium.

Plane analytics and the elements of infinitesimal calculus are also introduced during the tenth school year in the Realschule. The course in analytic geometry includes the derivation of the principal theorems by means of rectangular coordinates. The circle is studied and its equation is derived by the use of rectangular and by polar coordinates. The spiral of Archimedes is studied. The equation for the ellipse, the parabola, and the hyperbola are derived both in rectangular and polar coordinates. The ellipse, regarded as the projection of the circle, is also taken up.

Infinitesimal calculus.-The principles of the theory of limits and their application to the mensuration of the circle and of the surface and volume of the cylinder, cone, and sphere are studied. The limiting value of as z tends to zero and that of

sin z

The system of natural

when n increases without limit, are discussed. logarithms is presented. Continuity of functions is studied. The geometric representation of functions and derivatives receive attention. Formulas for the derivatives of the sum, difference, product, and quotient of functions are taught, and the derivatives of inverse functions are also presented. The subject of maxima and minima is taken up. Equations for tangents and normals at a given point of a curve are derived. Indefinite and definite integals are taken up.

Those pupils who elect this course in mathematics are usually preparing for the school of technology.

SPAIN. No pure mathematics is offered during the tenth school year. Both algebra and trigonometry are completed during the preceding year and physics is studied during the tenth year.

SWEDEN. The tenth school year is the second year of the Gymnasium. In the classical Gymnasium four hours a week, and in the Realgymnasium six hours a week are devoted to the study of mathematics.

Classical Gymnasium.-Involution and evolution and the use of tables of logarithms are taught. The pupils graph simple functions. The geometry of previous years is reviewed and numerous applied problems are solved.

Realgymnasium. Simple trigonometric computations for right and oblique triangles are made. The function concept is emphasized throughout the year. The work in trigonometry is a continuation of the theory of projections and the intersection of planes by planes and of solids by planes.

SWITZERLAND.-The tenth school year is the fourth year of the Gymnasium and the second year of the Realschule.

Gymnasium.-One and a half hours a week are devoted to the study of algebra. The course includes powers and roots, equations of the second degree involving one unknown and equations of higher degree reducible to equations with one unknown, imaginary and complex numbers.

One and a half hours are usually devoted to the study of geometry. The course includes the study of similarity of polygons, cyclometry, applications of algebra to geometry, harmonic division and transversals.

Plane trigonometry is introduced and the function concept is given a great deal of emphasis.

Realgymnasium.—Two hours a week are devoted to the study of algebra, and three and a half hours to geometry and trigonometry. The course includes all that is given in the Gymnasium, with the addition of dihedral and polyhedral angles, and a more intensive study of trigonometry and goniometry.

Realschule. The course is practically the same as in the Realgymnasium. The work in algebra is somewhat more intensive and includes some equations of the third degree, logarithms, arithmetical and geometrical progressions.

UNITED STATES.-There are two general plans for organizing the course in mathematics for the tenth school year.

The first is the older plan and is still in more general use, but the second is growing in popularity.

First plan: Half of the year is devoted to the study of algebra. The course includes the theory of exponents, radicals, the theory of quadratics, simultaneous quadratics, the binomial theorem for positive integral exponents. This is followed by a half year of plane geometry. The first two books are completed.

Second plan: The entire year is devoted to the subject of plane geometry, and the five books are completed.

SUMMARY OF THE TENTH YEAR'S WORK.

The tenth school year is the second year of the secondary school in the United States. In most of the schools the entire year is devoted to the study of plane geometry. In some of the schools half of the year is devoted to the study of algebra and the other half to the study of plane geometry.

In the schools of Europe the mathematics of the tenth school year is distinctly in advance of that in the United States. The difference in the mathematical courses in the two countries is most marked in the subjects of geometry, trigonometry, and drawing.

In most of the European schools the course in algebra is practically the same as in the first half of the third year in the secondary schools in the United States, that is, the course begins with the study of quadratic equations and continues through the progressions and the binomial theorem. In many of the European countries the subjects of logarithms, compound interest, and annuities are given a good deal of attention. The courses in geometry abroad usually include the study of congruent and similar figures, cyclometry, regular polyhedra, and the formulas for the surfaces and volumes of the common solids.

The subject of trigonometry is taught during the tenth school year in almost all of European countries. The extent of the course varies, but usually the entire subject of plane trigonometry is completed during this school year. In a few of the countries spherical trigonometry is also studied, and the knowledge thus gained is applied in the study of solid geometry and of mathematical geography.

Descriptive geometry is introduced in the French scientific lycée, in a few Realschulen and Oberrealschulen of Germany, and in some of the schools of Holland and Hungary. The study of geometric drawing is given more emphasis in France than in other countries. The elements of analytic geometry are introduced in a few of the Realgymnasia and Oberrealschulen of Germany and in some of the Realschulen of Russia. In practically all of the European countries, by the time a boy has completed the tenth school year, if he has pursued the scientific course, he has studied all the mathematics that is offered during the entire 12 school years in most of the schools of the United States.

XII. THE WORK IN MATHEMATICS IN THE ELEVENTH SCHOOL YEAR.

AUSTRIA. The eleventh school year is the last year of the Realschule and the seventh year of the Gymnasium and Realgymnasium.

Gymnasium.-Plane geometry is completed during the tenth school year. A good deal of emphasis is placed upon the idea of function. Spherical trigonometry is not taught in the Gymnasium. Surveying and mechanics receive a great deal of attention. Analytic geometry is introduced. The study is at first confined to rectangu