of simple equations with one or two unknowns, applied problems, and negative numbers.

The course in geometry includes the study of the parallel and perpendicular planes, the principal properties of the prism, cylinder, cone, and sphere, and the areas and volumes of these solids. The most difficult problems are omitted. An attempt is made to give the pupil some idea of elementary surveying. The final examinations which occur at the end of this year involve problems the solution of which demands some knowledge of arithmetic, algebra, and geometry.

RUSSIA. The eighth school year is the fifth year of the Realschule and of the Gymnasium.

In the Gymnasium four periods a week are devoted to mathematics. Two hours are usually given to algebra and two to geometry.

The course in algebra includes the solution of quadratics with one unknown, a discussion of the properties of the roots of quadratic equations, the factoring of the quadratic trinomial, the solution of simultaneous equations, and radical equations.

The course in geometry includes the measurement of straight lines and angles, the proportionality of segments, the similarity of triangles and polygons, the numerical relations between the sides of a triangle, elementary ideas of limit, the principal properties of the circle, areas of rectilinear figures and of circles, simple problems in construction, dihedral, and polyhedral angles.

In the Realschule three periods a week are devoted to algebra and three to geometry. The course in algebra is said to include the study of square root, irrational numbers, simple quadratics, irrational roots, the relation between roots and coefficients in quadratics, construction of squares from given roots, simultaneous quadratics, arithmetical and geometrical progressions, infinite series, and circulating decimals.

The course in geometry includes the measurement of angles, the study of proportional intercepts, the similarity of triangles and polygons. Inscribed and circumscribed triangles and quadrilaterals are studied and elementary ideas of limits are presented. The pupils learn how to compute the area of a circle. Simple constructions are made, and the principle of homogeneity is presented.

SPAIN.-The report does not give details for the work of this year, but the statement is made that algebra and trigonometry are taught, six periods a week being devoted to the subject of mathematics. The instruction in geometry is of an intuitive character.

SWEDEN.-The eighth school year is the fifth year of the Realschule. Equations of the first degree in one and two unknowns are studied. Two hours a week are devoted to arithmetic and two hours to geometry. Simple bookkeeping is usually included in the course in arithmetic. Square-root tables are extensively used. The opinion prevails that algebra should be introduced through the simple equation. No textbook is used in the introductory work in geometry. The regular plane figures and the circle are studied. The principal properties of the ellipse are considered and some attention is given to projections.

SWITZERLAND.-The eighth school year is usually the last year of the middle school. It is the second year of the Gymnasium.

The Gymnasium.—Half an hour a week is usually devoted to the study of arithmetic, two hours to algebra, two to geometry, and one to geometric drawing. In several of the Cantons no special instruction is given in the subject of geometric drawing. The courses in the Gymnasia vary in the different Cantons, but in general, they are somewhat as follows:

In arithmetic, simple and compound proportion, abridged multiplication and division, square root, and the computation of simple areas, partnership, alligation, and business transactions are studied.

The course in algebra includes the four operations and the solution of simple linear equations.

The course in geometry includes the fundamental ideas of the subject up to the equality of surfaces. The applications of the theorems are emphasized, and a good deal of attention is placed upon drawing and the simple constructions.

UNITED STATES.-The business applications of percentage and the mensuration of common surfaces and of solids include all that is usually offered in mathematics during the eighth school year in the United States. In some schools, a comprehensive review of the arithmetic of previous years is given. In the most progressive schools of the country no arithmetic is given during the eighth school year, or at least during the last half of the eighth school year, and the time is devoted to the study of algebra combined with arithmetic.

The New York State course of study outlines the following in arithmetic for the eighth school year:

Daily drill in rapid mental and written computations. This includes certain short methods in multiplication; the computation of simple interest by short methods; and drill in the use of proper business forms, such as invoices, bills, and account sales. The pupils are taught how to keep the simple accounts of daily life, of the home, and the farm. Stocks and bonds are considered. The various types of insurance are studied, and the subject of taxes is closely correlated with the subject of civics and is taught from the stand point of local conditions. The simple formulas common in various mechanical journals and in trade papers are used to introduce the study of algebra. The study of these formulas is followed by the solution of simple linear equations.

Throughout the eighth school year, especially, an attempt is made to correlate the work in mathematics with the instruction in manual and household arts and agriculture. The problems are largely of a practical nature, and much data is based on local conditions. In some of the schools, a great deal of attention is devoted to the subject of proportion. This is especially true in those schools where the subject of mensuration is emphasized, and the simple equation is not introduced early.

SUMMARY OF THE EIGHTH YEAR'S WORK.

In the United States the eighth school year is the last year of the elementary school. In most of the schools but little work in mathematics is attempted beyond the study of mensuration, proportion, and some of the applications of percentage. In an in creasing number of schools some work in algebra is being introduced during the eighth school year. The nature and extent of this work varies, but in general it may be said to consist chiefly of the simple equation and the evaluation of formulas. It is but rarely that a school is found in which any attention is given to the subject of geometry except such as is necessary in the study of mensuration.

In all European countries the course in mathematics is more extensive than in the United States. In no European country is less geometry or algebra offered during the eighth school year than in the United States, and in practically all the countries the work in both of these subjects is much more extensive than in this country. In Austria, Germany, and Switzerland the most emphasis in algebra is put upon the four operations, the solution of simple equations, and proportion. In a few of the countries the subject of factoring is also taught. This is especially true in certain schools of Austria, Roumania, and Holland. The course in algebra reported for Russia is more advanced than in any of the other countries, but it is probably safe to assume that only the most elementary ideas of some of the topics mentioned are presented to the pupils.

In practically all of the countries, except the United States, a good deal of emphasis is put upon the study of intuitive geometry during the eighth school year. Especial attention is devoted to the study of congruent and similar figures and to simple constructions. In a few of the countries the pantograph is used in connection with the

study of similar figures, and the terms sine, cosine, and tangent are introduced. This is notably true in France. In a few of the modern lycées of France logarithms are introduced. The subject of geometric drawing is given a good deal of attention in several of the countries, especially in Austria, Hungary, and in some of the Cantons of Switzerland. In a few of the schools of Austria some instruction is given in the subject of descriptive geometry. In practically all of the schools abroad the subjects of geometry and arithmetic are more closely related than in this country. Continued emphasis is put upon the function concept idea, especially in Austria, Germany, and Switzerland.

In most of the foreign countries less time is devoted to the study of arithmetic during the eighth school year than in the United States. The study of alligation is continued in some of the schools of Austria, Belgium, Switzerland, and Japan; and a good deal of emphasis is put upon the study of tests for divisibility.

It is customary in all of the European countries to teach algebra and geometry simultaneously. During the eighth school year the time is about evenly divided between these two subjects, from two to three hours a week being devoted to each. An attempt is not made to fuse the subjects, but the interrelations between them are kept constantly in mind, and the pupil is not permitted to forget his geometry while studying his algebra, or vice versa. Each subject is considered an instruction unit, but it is used whenever possible as a tool in the study of the other. By the time a European boy has completed his eighth school year, he is at least a full year in advance of the American boy in his knowledge of mathematics.

X. THE WORK IN MATHEMATICS IN THE NINTH SCHOOL YEAR.

AUSTRIA.-The ninth school year is the fifth year of the Gymnasium and of the

Realschule.

In the Realschule four hours a week are devoted to the study of mathematics. This time is about evenly distributed between arithmetic and geometry. These two subjects complement each other and form one instruction unit. Accurate observation and concise description are emphasized. Modeling is extensively used. There is a close correlation between plane and solid geometry, because of the extensive use of models. The theory of powers and roots is studied. Quadratics with one unknown are solved, and the pupils learn to graph the various types of quadratic equations. Only the simplest types of simultaneous quadratics are considered. Irrational, imaginary, and complex numbers are studied as far as is necessary for the solution of quadratic equations. The subject of logarithms is thoroughly studied. In geometry the work of the fourth class is continued and completed during the first semester. The course includes the study of proportional lines, similarity, and the computation of areas. During the second semester the subject of stereometry is introduced. The instruction begins with the study of solid angles. Tetrahedrons are studied, and surfaces and volumes are computed. A systematic study of descriptive geometry is made during this year. The course includes the study of straight lines and planes, vertical and horizontal projections. Oblique projections are used occasionally. Constructions are applied in the study of the regular pyramid and prism and of their shadows.

In the Realgymnasium two hours a week are devoted to descriptive geometry. No special hours are assigned for the subject of geometric drawing. With this exception, the course is practically the same in mathematics as in the Gymnasium. In the Gymnasium neither descriptive geometry nor geometric drawing is oblig atory. The arithmetic of previous years is extended and supplemented. Special

attention is given to the subject of powers and roots. Stereometry is introduced and emphasis is put upon functional thinking. Models are very extensively used.

BELGIUM.-The ninth school year is the last year of the middle school and the third year of the Athénée Royal.

Middle school.-A comprehensive review of the arithmetic of previous years is given. Powers and roots are studied, special attention being devoted to approximate roots. Compound interest is computed by the use of tables. Bonds, shares, savings banks, annuities, and insurance are studied. Tables are extensively used in all computations. Algebra. The course includes the study of the fundamental operations, the square and cube of a binomial, fractions, equations of the first degree involving two and more unknowns, negative numbers, and indeterminate equations.

Geometry. The work of previous years is reviewed, and proportionality and similarity are introduced. The relations between the sides of a right triangle are computed. The regular polygon, circle, and the sector are studied. Some elementary exercises are given in surveying and in leveling. The surfaces and volumes of polyhedra and of cones, cylinders, and spheres are computed and applied in solving practical problems.

During this year, girls are not required to study algebra. The course in arithmetic is the same for boys and for girls. The course in geometry is somewhat less extensive for girls than it is for boys. A study is made of parallels and of elementary theorems. The areas of rectangles, parallelograms, triangles, trapezoids, and circles are computed, and the formulas for the surface and volume of prisms, cylinders, pyramids, cones, and spheres are used without proof. From three to five hours a week are devoted to the subject of mathematics.

Athénée Royal.-In the classical schools three hours are devoted to mathematics, and in other types of schools four hours are devoted to the subject.

The course in the classical school includes a study of the changes which a qoutient and remainder undergo when the dividend and divisor, or one of them, are increased or decreased in a certain ratio. Tests for divisibility are studied. Checks by the casting out of nines and the elevens are applied in multiplication and division. Simple interest, bank discount, annuities, stocks and bonds, mixtures, alligation, negotiable paper, partnership, arithmetical and geometric progression, simple and compound proportion, and the computation of easy surfaces and volumes are studied during the year.

The course in the modern school is much the same as in the classical school, except that no attention is given to the study of arithmetical and geometrical progression, and more emphasis is put upon the study of congruency and simple constructions in geometry.

DENMARK.—The ninth school year is the last year of the intermediate school. At the close of the year a comprehensive examination is given, and the student must pass this satisfactorily before he is entitled to enter a higher grade. Seven hours a week are devoted to the subject of mathematics. Two hours are usually given to arithmetic, three to algebra, and two to geometry. After the work of the previous year has been thoroughly reviewed, the subjects of proportion, powers and roots, simple equations and quadratic equations with numerical coefficients are studied.

The course in geometry includes a review of the work of the previous year and a study of the similarity and congruency of triangles and polygons. Simple constructions are made. Provision is made for daily drill in mathematics.

ENGLAND. The lack of uniformity in the English school system, especially in

the upper years, makes any exact statement in regard to courses quite difficult.

In some of the schools a course similar to the following is offered for boys who do not wish to specialize in mathematics: Extraction of the square root by rule, the Pythagorean theorem, the study of circles, chords, arcs, tangents, and angles; the construction of circles from simple data, the construction of regular polygons, the

solution of quadratic equations with numerical coefficients, the simplifying of fractions, and the solution of applied problems.

The boys who pursue this course are not expected to continue the study of mathematics after leaving the school. The majority of them do not enter the university. The boys who expect to enter the universities and who are preparing at any of the great schools, such as Eton, Rugby, Harrow, and Winchester, spend from three to seven hours a week in the study of mathematics. By the time a boy is 15 or 16 years old he has usually completed the study of the five books of Euclid and of algebra to the progressions.

FINLAND.-Boys who expect to pursue their education beyond the elementary school usually enter the lycée or the secondary school. The lycée prepares directly for the university. The classical and modern lycée are somewhat similar to the German Gymnasium and Realschule. Both algebra and geometry are taught. Emphasis is put upon the solution of simple and quadratic equations. Solid geometry is briefly studied. The development of the intuition receives a good deal of attention. Further details of the course are not available.

FRANCE. The ninth school year is the fifth year of the lycée. It is the first year of the second cycle. The second cycle lasts two years and has four divisions:

A. Latin-Greek (classical).

B. Latin-Modern languages.

C. Latin-Science.

D. Science-Modern languages.

Course in A and B.-Two hours a week are devoted to the subject of mathematics. The work in algebra includes a review of the work of the previous year. The four operations are performed with positive and negative numbers. Problems involving uniform movement are solved. Inequalities of the first degree are studied. Variations of the expression ax+b are considered. Graphs are introduced, and the pupils represent the variations of x2 and of

1 x

Solid geometry.-Dihedral angles and perpendicular and parallel planes are studied. Polyhedral angles are considered; and formulas for the surface and volume of prisms, pyramids, cylinders, cones, and spheres are developed. The course does not include geometric drawing.

Course in C and D.-Five hours a week are devoted to the subject of mathematics. The work is more intensive and more extensive than in courses A and B. The four operations with positive and negative numbers, the solution of equations of the first degree involving one and more unknowns, and inequalities of the first degree are studied. The variation of the expression ax+b is considered and represented graphically. Equations of the second degree in one unknown are solved, but no equations involving imaginaries are considered. The relations between roots and coefficients are studied. The quadratic trinomial and inequalities of the second degree are introduced. A good deal of emphasis is put upon the subject of graphs. The variations of the exax+b pression are considered. The notion of derivatives is applied to simple numera'x+b ical problems and to functions previously studied. Arithmetical and geometric progressions are introduced. Four-place logarithms are used, and the subject of compound interest is studied.

The course in geometry includes a systematic study of lines, angles, parallels, perpendiculars, triangles, quadrilaterals, polygons, and circles. The theorems for the congruency and similarity of triangles are especially considered, and the terms sine, cosine, tangent, and cotangent for angles from zero to 180° are introduced. The construction of mean and fourth proportionals and the harmonic division of a line are considered. The areas of triangles, polygons, and circles are computed. Ele