mentary notions of surveying are introduced. The pupils learn to use the surveyor's chain and the square.

In the classes in drawing, the ruler, compasses, square, and protractor are used. The simple geometric constructions are made.

The course in geometric drawing includes the making of designs particularly for tiles, parquets, and mosaics. Many of the drawings are colored.

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

In the Gymnasium two hours a week are usually devoted to the study of algebra and three to geometry. The courses vary somewhat in the different States, but the work in algebra usually includes the solution of simple equations with one or more unknowns, graphs, the study of simple functions, and the solution of simple quadratic equations. In some of the States the course includes also the use of five-place logarithms and the slide rule, and the study of arithmetical and geometric progressions, compound interest, and annuities. The course in geometry usually includes the study of similar figures, regular polygons, and cyclometry.

In most of the Realgymnasia and Realschulen, three hours a week are devoted to algebra and three to geometry. The work is more intensive than in the Gymnasium, especially in the subject of logarithms and quadratic equations. Elementary trigonometry and solid geometry are sometimes introduced, and oblique and parallel projections are considered.

The course in the Oberrealschule is more advanced than in the other types of schools. In general it may be said to be an intensive study of the work that is offered in the Gymnasium, with the addition of the following: Trigonometric functions, and goniometry are frequently introduced with the study of similar figures. Solid geometry is studied. Euler's theorem is considered. The study of regular polyhedra is introduced, and numerous exercises involving numerical computations are solved.

HOLLAND. The ninth school year is the third year of the middle school and of the Gymnasium.

Seven periods a week are devoted to the study of mathematics in the middle schools. The previous work in radicals is reviewed and extended. Fractional negative exponents are studied. Logarithms, arithmetical and geometrical progression, and compound interest are introduced. Equations of the first and higher degrees involving one or more unknowns are solved. The course in geometry includes the study of circles, regular polygons, and goniometry.

The Gymnasium.-All types of Gymnasia offer the same course in mathematics. One hour a week is usually devoted to the subject of algebra and two to geometry. The course in algebra includes the study of equations of the first and second degrees with one or more unknowns and the subject of radicals. Details of the course in geometry are not available.

HUNGARY.-The ninth school year is the fifth year of the Gymnasium, Bügerschule, and Realschule.

Three hours a week are devoted to mathematics in the Gymnasium. Systematic instruction in geometry begins during this year, and a good deal of attention is devoted to the study of models. The pupils are required to make numerous indoor and out-ofdoor measurements and estimates. The axioms, postulates, and the principal theorems of plane geometry are studied. The course in algebra includes the solution of equations of the first degree with one or more unknowns, square root, cube root, radicals, and quadratic equations of the types used in geometry.

Realschule.-Five hours are devoted to the study of mathematics. No geometric drawing is given in either Gymnasium or Realschule. The course is practically the same as in the Gymnasium, but the work is more intensive. Some of the fundament theorems of solid geometry are introduced.

Bürgerschule.-Four hours a week are devoted to the study of mathematics. The course includes the four fundamental operations with positive and negative numbers, fractions, simple linear equations in one unknown, the chief theorems of planimetry, the study of congruent and similar plane figures, the construction of plane figures, and the study of perimeters and areas.

ITALY.-The ninth school year is the last year of the Ginnasio and the third year of the modern school. Three hours a week are devoted to the study of practical arithmetic and two hours to geometry. The first book of Euclid is studied. Some attention is devoted to the study of the simple solids and to the proportionality of magnitudes. Equations of the first degree are solved. Four periods a week are usually devoted to mathematics.

JAPAN.-The ninth school year is the last year of the higher primary and the third year of the middle school.

The higher primary.-Four hours a week are devoted to the study of mathematics. The lessons of the previous years are reviewed and supplemented. There is a repetition of the mensuration previously learned, to which are added the mensuration of the pyramid, circular cone, frustum of a pyramid, frustum of a circular cone, expression of square root and cube root and a review of the four rules, proportion and percentage, together with arithmetical and geometric progression.

Rules for finding the volume of pyramids and circular cones are given dogmatically and verified by actual measurement. Simple bookkeeping is sometimes taught.

The middle school.-Four hours a week are devoted to the study of mathematics. The study of equations is continued, and linear simultaneous equations are solved. Formulas relating to the distributive law are considered, factoring, the greatest common factor, and lowest common multiple are studied. The reduction of fractional expressions and the four fundamental operations with fractions are included in the course. Quadratic equations with one unknown quantity, equations with one unknown quantity reducible to quadratics, and simultaneous equations containing quadratic equations are presented.

The course in geometry includes the study of straight lines, triangles, parallelograms, circles, chords, tangents, and segments. Two hours a week are devoted to this work.

ROUMANIA.-The ninth school year is the first year of the lycée. Six hours a week are devoted to the study of mathematics. The four operations with common and decimal fractions are explained. The use of tests for divisibility is continued. Prime numbers are studied. The idea of the limit is introduced. Square and cube root are studied. The aim is to cultivate mathematical reasoning. The work is more intensive than in previous years.

Algebra.-Positive and negative numbers, algebraic fractions and radicals are introduced. Equations of the first degree are solved, and some attention is devoted to the study of reciprocal equations. The quadratic trinomial is considered. Arithmetical and geometrical progression, logarithms, compound interest, and annuities are taken up.

The course in geometry includes the study of angles and triangles, quadrilaterals, circles, and proportion. Numerous exercises are solved.

RUSSIA. The ninth school year is the sixth year of the Realschule and of the Gymnasium.

In the Gymnasium four hours a week are devoted to the subject of mathematics. The work in algebra includes the study of progressions, logarithms, and compound interest. The course in geometry includes the study of regular polyhedra, the formulas for the area and volume of the prism, cylinder, pyramid, cone, and sphere. A good deal of attention is devoted to estimates.

In the Realschule two periods are devoted to algebra, two to geometry, and two to trigonometry. Fractional, negative, and zero exponents are introduced. Irra

tional expressions are considered. Logarithms, exponential equations, combinations, the binomial theorem, continued fractions and their application to square root and to logarithms, compound interest, and annuities are studied.

The course in geometry includes the study of the relative positions of straight lines and planes in space, the chief characteristics of plane and solid angles, the regular polyhedra, and the study of the formulas for the surfaces and volumes of the regular solids.

In trigonometry, the sine, cosine, tangent, and cotangent for both acute and obtuse angles are studied. The limiting values of these functions are considered and the logarithmic and trigonometric functions are used. The elementary formulas of plane trigonometry and the formulas for the solution of right and oblique triangles are developed. The areas of triangles are computed by the use of the trigonometric formula, and trigonometry is applied to the solution of geometric problems and exercises.

SWEDEN.-The ninth school year is the last year of the Realschule and the first year of the Gymnasium.

In the Realschule square root and its application to geometric problems, the computation of compound interest by means of tables, graphs, and simple bookkeeping are taught. Two hours a week are devoted to the study of arithmetic.

The work in geometry includes the drawing and making of models, simple projections, the use of the compasses, ruler, and protractor, and the making of simple constructions. One hour a week is devoted to geometry.

Gymnasium.-Five hours a week are devoted to mathematics in the classical gymnasium, and seven hours a week in the realgymnasium. The algebra of the Realschule is reviewed, and square root and quadratic equations are studied. Simple functions are emphasized. The geometry of the Realschule is reviewed, and numerous applications are made. Linear drawing is given considerable attention, and the principles of the theory of projection are studied. Special attention is directed to the study of the intersection of plane and solid figures.

SWITZERLAND.-The ninth school year is the third year of the Gymnasium and the first year of the Realschule. The course is the same in the Classical and the Realgymnasium. Two hours a week are devoted to algebra, two to geometry, and in some Cantons two hours a week are devoted to geometric drawing in the Realgymnasia.

The course in algebra includes the study of logarithms, arithmetical and geometrical progression, compound interest and annuities, the four operations with fractional exponents, and the study of functions.

The course in geometry includes the study of transversals, triangles, quadrilaterals, circles, construction problems, the theorems of congruency and similarity, and the study of surfaces and volumes.

Realschule.-One hour a week is devoted to the study of arithmetic, two to algebra, three to geometry, and two to geometric drawing. The course is practically the same as above except that a more intensive study is made. More attention is given to abridged multiplication and division than is the case in the Gymnasium. UNITED STATES.-The course in mathematics varies somewhat for the ninth school year, but in general it may be stated as follows:

The four fundamental operations with rational integral expressions, factoring, highest common factor and lowest common multiple, fractions, reduction, the fundamental operations with fractions, linear equations with one unknown, applied problems, simple and simultaneous equations with one and more variables, graphic solutions, involution, the method of extracting square and in some cases cube root, fractional and negative exponents with proofs of fundamental laws, radicals, irrational expressions and equations, problems, the solution of quadratic equations of the type ax2+bx+c=0 by factoring.

SUMMARY OF THE NINTH YEAR'S WORK.

The ninth school year is usually the first year of the secondary school in the United States. In the majority of secondary schools the subject of algebra is completed up to quadratic equations. In a few of the schools less work is attempted in algebra and more attention is devoted to the subject of intuitional geometry. In such schools, emphasis is put upon the use of the simple geometrical instruments, the making of easy constructions, measurements, and simple designs.

In all European schools both algebra and geometry are taught during the ninth school year. In most of the countries the time is divided evenly between these two subjects, from two to three hours a week being devoted to each. Here, as in the preceding year, the subjects supplement each other, and no attempt is made to fuse them. The relations between the two subjects are emphasized much more extensively abroad than in the United States. In most of the schools of Europe the distinction between plane and solid geometry is less marked than in the schools of the United States. This is, in part at least, due to the fact that models and drawings are very extensively used abroad. During the ninth school year the work in algebra in most of the European schools is but slightly more advanced than in the best schools of the United States. In some of the schools of Austria, France, Denmark, Holland, and Hungary the course is somewhat more advanced than in the United States. In these schools logarithms, proportion, and quadratics are studied. . The course in the Realschulen of Russia is said to be very advanced.

In practically all of the European countries geometry is studied during the ninth school year. Especial emphasis is put upon the study of proportional lines, congruent and similar figures, areas, and volumes. In all of the schools models are extensively used. In Belgium, France, Germany, and Holland the study of similar figures is supplemented by elementary exercises in surveying, and in some of the French schools the terms sine, cosine, tangent, and cotangent are introduced and used.

Descriptive geometry is introduced in some of the Realschulen and Realgymnasia of Austria. In France, Sweden, and in the Realgymnasia of Switzerland some attention is devoted to the subject of geometric drawing. In most of the countries continued emphasis is put upon the idea of functions. This is especially true in Austria, France, Germany, Sweden, and Switzerland. The subject of logarithms is introduced in most of the countries, and in some of the schools of Germany the pupils become familiar with the slide rule.

In a few of the countries the study of arithmetic is continued during the ninth school year. Especial emphasis is put upon the use of tables for finding interest, and upon abridged processes.

Arithmetic, algebra, and geometry are more closely related in the schools abroad than in the United States. Each subject is considered as an instruction unit, but it is closely correlated with the others. When a European boy has completed his ninth school year he has a mastery of all the topics in algebra that the American boy haş studied, and in addition to these he is familiar with practically all of the more important theorems of plane and solid geometry. In some of the schools of Finland, Germany, and Russia the elements of trigonometry are introduced.

XI. THE WORK IN MATHEMATICS IN THE TENTH SCHOOL YEAR.

AUSTRIA. The tenth school year is the sixth year of the Gymnasium, Realschule, and Realgymnasium.

Realschule. The course in algebra includes the study of logarithms and exponential equations of the simplest kind, arithmetical and geometrical progressions, annuities, and compound interest.

Goniometry and plane and spherical trigonometry have an important place in the curriculum, on account of their practical value. The course in trigonometry may be said to dominate the work of this school year in the Realschule. The functions are defined and developed for acute and obtuse angles and are graphically represented in the unit circle. All the fundamental formulas are developed and used in the solution of acute and oblique triangles. Logarithmic tables and the natural functions are used. The spherical triangle and the chief formulas for its surface are studied. These formulas are applied to the solution of certain problems in stereometry.

The instruction in geometric drawing in the Realschule and in the Realgymnasium includes an extensive study of vertical projections of plane figures, the shadow of the circle projected on a plane by parallel and by oblique rays, the derivation of the properties of the ellipse conceived as the projection of the circle, the representation of cylinders and cones of revolution, tangential planes, cones and cylinders, shadow constructions, and the study of the sphere. Three hours a week are devoted to the subject of geometric drawing.

Gymnasium.-Equations of the second and of higher degrees that can be reduced to quadratic equations are solved. Surds, imaginaries, and complex numbers, graphs, and the nature of roots of quadratics are considered.

Geometry. A full year is devoted to the study of solid geometry, a great deal of emphasis being put upon drawing. The pupils draw vertical and horizontal projections of all the simple bodies.

Goniometry and trigonometry.-The course is not so extensive as in the Realschule, and the slide rule is not much used in the instruction. Five-place logarithmic tables are almost always used.

BELGIUM.-The tenth school year is the fourth year of the Athénée Royal. In the classical course three to four hours a week, and in the modern course four hours a week, are devoted to the study of mathematics.

Classical course.-The arithmetic of previous years is reviewed. The greatest common divisor by successive division is studied. Factoring and greatest common divisor and lowest common multiple by factoring receive a good deal of attention. The course in algebra includes the transformation of equations, the study of proportion, linear equations with one unknown, applied problems, the evaluation of formulas, and alligation.

The work in geometry includes the study of the first book of Euclid. Numerous constructions are made.

Modern course. The work in arithmetic is the same as in the classical course, with the addition of the study of tests for divisibility.

The course in algebra includes all that is given in the classical course, with the addition of negative expressions, the square and cube of binomials, the four fundamental operations, the remainder theorem, the four operations with fractions, the reduction of fractions, squares and the square roots of numbers, and of algebraic expressions, radicals, and simple quadratics.

The course in geometry includes a complete review of all the work of preceding years and a study of the circle, the mensuration of quadrilaterals, the study of proportional lines and similar figures, and the calculation of the elements of the triangle. DENMARK.-The tenth school year is the first and only year of the Realklasse and the first year of the Gymnasium.

Realklasse. This class is for those who wish an additional year, but are not going to the university.

The work in arithmetic includes the study of business forms, the use of interest, and logarithmic tables.

The course in algebra includes the general solution of quadratic equations. The rules for the sum and the product of the roots, the solution of easy equations reducible

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