FRANCE.-In some schools the higher primary lasts but six years. In others there is an additional year which corresponds to our seventh school year. The seventh school year is also the third year of the first cycle of the lycée.

Higher primary-In view of the examinations, there is a comprehensive review of the mathematics of previous years.

Geometry. The pupils are taught how to compute the areas of rectangles, parallelograms, triangles, trapezoids, polygons, circles, sectors, and segments of circles. Comparisons of the areas of similar polygons are made, and the formulas for the volumes of prisms, pyramids, and the frustrum of pyramids are developed. The volumes of similar polyhedra are compared. The formula for the volume of the cylinder, cone, and frustrum of the cone are developed. The rules for finding the surface and volume of a sphere and the surface of a zone are taught. The fundamental ideas of descriptive geometry and the elementary notions of rotation are presented.

Three hours a week are devoted to the subject of geometric drawing. The course is a review and a slight extension of the course of the previous year.

Classical lycée.-Two hours a week are devoted to the subject of arithmetic. The course includes factoring, tests for divisibility, prime numbers, proportion, and the square root.

Geometry. The ruler, square, compasses, and protractor are used. Straight lines, angles, triangles, quadrilaterals, circles, cords, arcs, and tangents are studied. The congruency of triangles, and the theorems for the sum of the angles of a triangle and of a polygon are developed intuitively. Elementary constructions involving straight lines and circles are made and numerous angles are measured. No geometric drawing is taught.

Modern language lycée.-Four hours a week are devoted to mathematics and one to geometric drawing.

The course in arithmetic includes the study of fractions, the extraction of the square root, progressions, and the commercial applications of percentage.

Geometry. The pupils are taught how to divide a line into parts having a given ratio. Similar triangles are studied, and the definitions of sine, cosine, and tangent are taught. The pantograph is extensively used in connection with the study of similar polygons. Fourth proportionals and mean proportionals are constructed. Regular polygons are studied, and the formulas for the area of a rectangle, parallelogram, triangle, trapezoid, and polygon are developed. The areas of similar polygons are compared, and the areas of circles are computed. Some simple curves, such as the cissoid and the conchoid, are constructed. The course in geometric drawing is practically the same as in the sixth school year, with the addition of pen tracing. GERMANY. The seventh school year is the fourth year of the Bürgerschule, Gymnasium and Realgymnasium, and Oberrealschule.

The course in the Volksschule includes simple interest, profit and loss, commercial discount, averages, and alligation. In geometry, right-angle triangles, quadrilaterals, circles, cubes, prisms, pyramids, and cones are studied. A good deal of attention is devoted to measurements and to drawing. The drawing board is extensively used. Gymnasium and Realgymnasium.—One-third of the time is given to arithmetic and almost two-thirds of the time to geometry. Only a small part of the time is given to algebra.

The course in arithmetic includes interest, bankruptcy, and alligation. Addition, subtraction, multiplication, and division of algebraic expressions and the fundamental properties of equations of the first degree are taught. Instruction in geometry is of a propedeutic nature and includes the construction of plane figures, and the study of the congruency of triangles, the properties of the circle, chords, and angles. Some attention is devoted to geometric drawing.

Realschule and Oberrealschule.-Two hours are usually devoted to arithmetic and algebra and two to geometry. The course in arithmetic includes practical business

discussed without actual computations being made. Ratio and simple proportion are studied. The equation and the unknown quantity are used in proportion. The pupils are required to memorize the squares of numbers to 25. Square root is taught and is applied to numerous problems. The various tables of denominate numbers are reviewed, and numerous problems are based upon them. The work in mensuration includes the study of the surfaces of the parallelogram, rectangle, triangle, circle, and trapezoid; and the volume of the cube, sphere, cylinder, pyramid, and cone. No formal geometric demonstrations are attempted.

In some schools the topics of commercial discount, commission, and taxes are also studied during the seventh school year. Within recent years a great deal of attention is being devoted to the study of banks and banking, the saving and investment of money, and loans. In many schools it is thought profitable to organize school banks, to elect officers, and to carry on a regular banking business, usually with imitation money.

SUMMARY OF THE SEVENTH YEAR'S WORK.

In all of the European countries the course in mathematics is decidedly more advanced than in the United States.

In practically all of the countries abroad the subject of intuitional or observational geometry is emphasized during the seventh school year. This is the case in a few schools of the United States, but it is the exception rather than the rule. In some schools abroad the most elementary ideas of trigonometry are introduced in connec tion with the study of similar figures. The pupils abroad receive much training in constructive work, drawing to scale, reducing and enlarging figures. In some of the schools the pantograph and the planimeter are used. Squared paper is extensively used in England and to some extent in a few other countries. In Austria, France, Germany, Hungary, Italy, and Sweden especial attention is devoted to instruction in the elements of geometric drawing.

Geometry and arithmetic are very closely correlated in the instruction. In England more emphasis is put upon the geometric locus than in any other country during this school year.

In the United States more emphasis is probably put upon a few of the applications of percentage, but several topics in arithmetic not given much attention here are introduced abroad. Alligation is taught in few American schools, but the pupils in some of the schools of Germany, Hungary, and Switzerland study the subject. Recurring decimals, cube root, and the check for multiplication and division by casting out the nines, and the functional concept are more extensively taught abroad than in the United States. The function concept receives marked and growing emphasis in Austria, Germany, France, Hungary, and Switzerland.

It is the exception in the United States for any algebra beyond the simple equation and the evaluation of easy formulas to be introduced into the seventh school year. In practically all of the European countries, on the other hand, the pupils learn to solve simple linear equations containing one unknown, to evaluate the formulas of mensuration and percentage, and to perform the four fundamental operations with simple algebraic expressions. In a few countries some easy factoring and graph work are introduced.

IX. THE WORK IN MATHEMATICS IN THE EIGHTH SCHOOL YEAR.

AUSTRIA.-The eighth school year is the last year of the Bürgerschule and the fourth year of the Gymnasium, Realgymnasium, and Realschule.

Bürgerschule. The course in arithmetic includes powers and roots, ratio and proportion, simple and compound interest, partnership, alligation, and simple bookkeeping.

The four operations with algebraic expressions and simple equations of the first degree with one unknown are taught.

The course in geometry includes the intuitive study of the more important theorems of plane and solid geometry. The fundamental ideas of the ellipse and parabola are also presented. No distinction between plane and solid geometry is made, because much of the study is based on models. Geometry and arithmetic are closely correlated, and the function idea is emphasized.

The course in geometric drawing includes the representation of bodies in outline, transverse and longitudinal lines and triangles. Some attention is given to free-hand drawing and to designing. The course for girls is much the same as for boys, but the geometric drawing includes the construction of various designs for use in hand work. Realschule. The course in the Realschule begins with a thorough review of the fundamental operations in arithmetic. The principles underlying these operations are explained, and numerous practical problems are presented. Interest, alligation, proportion, and graphs are studied. Functional thinking is emphasized in the teaching of algebra by showing how the expression varies in value with different values of the unknown. Transformation of algebraic expressions and the solution of simple linear equations are taught. The pupils learn how to factor simple expressions according to type by both factoring and the Euclidean methods. Pure quadratic equations are studied as far as is necessary for the purposes of plane geometry.

In the course in geometry the theorems dealing with congruency and similarity are especially emphasized, and a good deal of attention is devoted to construction problems and to problems involving numerical computations. The time allotted to geometry in this class is generally considered inadequate.

Gymnasium. The study of intuitional geometry ends with the preceding school year, and a more systematic study of geometry begins during this year. The idea of function is not considered as a separate topic, but it permeates the whole course. Linear functions are considered in the fourth and fifth years of the Gymnasium, and graphs are used to supplement the idea. The course in geometry includes the study of straight lines, angles, parallels, triangles, and quadrilaterals. The congruency theorems and symmetry of plane figures are studied, and the study of regular polygons and circles is an important part of the year's work. Inscribed and circumscribed figures, tangents, chords, central, and inscribed angles are studied. Algebra and geometry are closely correlated, especially in the mensuration of plane figures. The geometry is a mixture of the Euclidean and post-Euclidean geometry. Pencils of rays and other modern concepts are studied.

Realgymnasium and Realschule.-The study of descriptive geometry is introduced through the theory of projections. The pupils learn to represent by drawings the various conic sections and to construct tangents to and from a point outside of a cone. Projectional drawings are made of simple bodies in various positions. Some simple shadow constructions are made. A study is made of the intersection of projecting planes. In the fourth-year class of the Realgymnasium there are no separate hours for geometric drawing.

BELGIUM.—The eighth school year is the second year of the middle school and of the Athénée Royal.

In the middle school the boys devote four hours and the girls three hours a week to mathematics. The course for girls does not include algebra, and the geometry for girls is the same as for boys in the previous school year, with the addition of some elementary constructions.

Course in arithmetic. The work of the previous year is thoroughly reviewed and the subjects of factoring and the highest common factor and the lowest common multiple by means of factoring are presented. Drill on the reduction of common to decimal fractions is continued. Discount, proportion, averages, mixtures, alligation, and aliquot parts are studied. Numerous oral exercises are given daily.

Algebra. The formulas of interest and of mixtures are studied, and the pupils learn to evaluate formulas for any letter and to translate formulas into words. Simple linear equations and ratio and proportion are studied.

Geometry. The chief aim of the work in geometry is to familiarize the pupil with the principal properties of triangles, quadrilaterals, and circles. Numerous problems involving the computation of the various parts of a triangle are given.

Athénée Royal.-In both the classic and modern schools three hours a week are devoted to the study of mathematics, and the course is practically the same in both types of schools. The course includes the rationalization of the rules for the four processes with integers, common and decimal fractions. The tests of divisibility by 2, 3, 4, 5, and 9 are presented. Checks are extensively used. Simple interest, commercial discount, mixtures, and alligation are studied. Problems in proportion are solved by the use of unitary analysis. An intensive study of the metric system is made.

DENMARK.—The eighth school year is the third year of the intermediate school. Six periods a week are devoted to the subject of mathematics. Provision is made for daily drill in oral arithmetic. The subjects of interest, partnership, profit and loss, simple bookkeeping, and proportion are taught. The tests for divisibility by 2, 3, 4, 5, 9, and 11 are presented. Simple areas and volumes are studied.

The subject of the geometry is introduced before algebra. The study is of an intuitive nature, and two hours a week are devoted to it. Accurate drawing is emphasized, and special attention is devoted to the construction and mensuration of circles. The Pythagorean theorem is intuitively established. Simple areas and volumes are computed, and numerous drawings and models are made.

ENGLAND. The subjects of rates and taxes, insurance, bankruptcy, and dividends are studied. Interest formulas are derived and used. Averages and proportion are given some attention. The subject of mensuration is treated largely from a practical point of view and includes numerous problems of carpeting, papering, and painting. Generalized arithmetic is introduced. Graphs are used to illuminate the subjects of proportion and interest, and squared paper is extensively used in the computation of areas. The investment of savings and the various methods of transmitting money are studied, and pupils become familiar with business practice. Simple lessons on the formation of companies are given, and the terms capital, stock, shares, shareholder, dividends, and investments are given a practical setting. Problems in measuring include areas and easy volumes. The course in geometry includes the study of the properties of triangles, the construction of triangles and ellipses, and the construction of simple geometric figures and designs. Ratio and proportion are studied. The Pythagorean theorem is discovered intuitively.

FINLAND.-The eighth school year is the fourth year of the lycée. The study of algebra is continued, and geometry is introduced.

FRANCE.-The eighth school year in France is the fourth year of the lycée. It is the last year of the first cycle.

In the classical lycée three hours a week are devoted to the study of mathematics. The work in arithmetic involves a study of the metric system and of proportion. In algebra, the pupils study the four operations with monomials and binomials. Negative numbers are introduced. Numerical equations of the first degree in one and two unknowns are solved, and inequalities of the first degree in one unknown are introduced.

Geometry. The work of the previous year is thoroughly reviewed. The study of proportion is introduced. The theorems involving the similarity of triangles and of polygons and of homogeneity are studied and the terms sine, cosine, tangent and cotangent are introduced. In the study of similarity, the pantograph is extensively used. The relations between the sides of a right triangle are computed. The properties of the circle, and of tangents, chords, and secants are studied. The formulas for

the area of the principal plane figures are developed, and the areas of two similar polygons are compared. No course in geometric drawing is offered in the classical lycée.

The modern lycée.-Four hours a week are devoted to the subject of mathematics. The course in algebra includes all that is given in the classical lycée, and in addition to this equations of the second degree involving one and two unknowns, graphs, logarithms, and antilogarithms to four decimal places, and compound interest are studied.

The course in geometry includes the study of planes and of straight lines in space. The terms polyhedral angles, prism, and pyramid are introduced, and the subject of projections is briefly studied. The formulas for the surfaces and volumes of prisms, pyramids, cylinders, and cones are deduced. The formulas for the surface and volume of the sphere are learned without proof. The pupils learn to draw simple designs and the elementary notions of surveying and of leveling are introduced. In geometric drawing, shadow drawings are given a great deal of attention and some of the drawings are colored.

GERMANY.-The eighth school year is the last year of the Volksschule and the fifth year of the Bürgerschule, Gymnasium, Realgymnasium, and Oberrealschule.

Volksschule. The course in arithmetic includes the study of insurance and of negotiable paper, and a review of the work of previous years, with special emphasis upon business computations. In some of the States, series and the theory of the extraction of roots are taught, but these are being eliminated.

The course in geometry includes the mensuration of circles, triangles, quadrilaterals, and ellipses. The cone and the sphere are studied. A good deal of practice is given in drawing, especially in drawing to scale. In some of the States the pupils make geometric designs and ornaments.

Gymnasium.—The course in algebra includes the four operations with algebraic expressions, the evaluation and transformation of formulas, and the solution of simple linear equations.

The course in geometry includes the study of lines, angles, and parallel lines, the congruency of triangles, quadrilaterals, circles, inscribed and central angles, loci, and the solution of simple problems by analysis.

Realgymnasium.-Four hours a week are devoted to the subject of mathematics. The course in algebra includes the four fundamental operations and the transformation of simple algebraic expressions. The course in geometry includes the study of straight lines, angles, triangles, and polygons.

The course in the Realschule is practically the same as in the Oberrealschule. Equations of the first degree with more than one unknown are studied. The pupils learn to represent graphically the multiplication tables and certain statistical data and to graph functions of the first and second degrees. Involution and evolution are taught and simple quadratics with one unknown are presented. Two periods a week are devoted to algebra.

Geometry. The course in geometry includes the study of the circle and the regular polygon. Numerous algebraic-geometric problems involving constructions are given. The reciprocal dependence of angles and sides of triangles is emphasized. The study of triangles prepares for the study of trigonometry. The important properties of prisms, cylinders, cones, polygons, and the frustums of polygons and cones are studied. These solids are presented in parallel perspective, and by orthogonal projections. The surfaces and volumes of the solids enumerated above are computed. HOLLAND.-The eighth school year is the second year of the middle school and of the Gymnasium. In the middle school, six periods of 50 minutes each are devoted to the study of mathematics.

The course in arithmetic includes the study of proportion and its applications, quare root, and approximate computations.