Gymnasium and Realgymnasium.-A comprehensive survey of the arithmetic of previous years is made. Rules are studied both in words and in letters, and simple transformations of formulas are made. The pupils are taught the use of parenthesis, and how to make substitutions in formulas and in equations. The ideas of negative number are presented.

The course in geometry includes the computation of simple surfaces, and the volume of the right prism, cylinder, pyramid, and cone. The pupils measure numerous objects both in and out of doors. The Pythagorean theorem is presented and is applied to plane and solid figures. Formulas for the surface and volume of the sphere are taught without proof. A good deal of attention is given to simular figures, and the idea is emphasized by means of reductions and enlargements. The pupils draw many figures to scale. The instruction in arithmetic and geometry is very closely related. The pupils are taught the graphic representation of the four fundamental operations and of (a+b)2, (a−b)2, (a+b), (a−b), (a+b)3, and (a-b)3. Abbreviated and approximate computations are given a good deal of attention, and square and cube root are presented. The pupils are taught to estimate their results and to check these estimates by measuring and weighing. Functional thinking receives continual emphasis. A few of the simplest equations are studied. Geometric drawing may be studied, but it is not obligatory. In some of the Realgymnasia it is required.

Realschule.-Three hours a week are usually devoted to mathematics and two to geometric drawing. The drawing is usually taught by the mathematics teacher, and this is considered a decided advantage. Arithmetic is always taught in close connection with geometry.

Approximate computation of decimals is presented and applied to the finding of surfaces and volumes. The pupils make approximate measurements of various objects, distances, and heights, and use the data in problems. General arithmetic is taught, and a summary of the work of previous school years is given. Rules are studied both in words and in letters, and the pupils are taught to generalize rules whenever it is possible. Simple transformations and checks are given a good deal of attention. The terms "coefficient," "powers," and "exponent" are taught. The pupils learn the rules for the square and cube of a binomial, and they represent graphically (a+b)2 and (a+b)3. Graphic representations of roots are also made. The idea of the negative number is presented, and the four operations with algebraic numbers are taught. The course in geometry is quite similar to the course in the Gymnasium. The pupils learn the fundamental laws regarding areas and volumes. The functional idea is especially emphasized during this school year. Figures are usually drawn to scale, and results are frequently checked by drawing. When solids are involved, results are usually checked by weighing. The course in geometric drawing relates principally to the mensuration and transformation of areas. Some attention is given to decorative forms using circles or arcs.

BELGIUM. The seventh school year is the first year of the middle school and of the Athénée Royal.

Course for the middle school.-The four fundamental operations with integers, common and decimal fractions are explained and extensively drilled upon. The pupils are taught the tests of divisibility for 2, 3, 4, 5, 8, 9, 11, 25, and 125. The casting out of nines is used as a check for multiplication and division. The greatest common divisor of two numbers is found by successive divisions. The changes effected by adding, subtracting, multiplying, or dividing both terms of a common fraction by the same number are studied. The subject of legal weights and measures is given a good deal of attention. Numerous problems involving the rule of three, simple interest, profit and loss, and commission are solved by means of unitary analysis.

Geometry. The fundamental concepts of geometry are presented and the pupils are led to discover the conditions under which triangles are congruent. The theorem

for the sum of the angles of a triangle is also developed. The principal properties of triangles and quadrilaterals are studied.

Athénée Royal.-Practically the same course is offered in both the classical and the modern-language schools. The fundamental operations with integers, common and decimal fractions are explained and drilled upon. Tests of divisibility for 2, 3, 4, 5, 6, 9, and 11 are taught without proof. The method of reducing a common to a decimal fraction and the various principles of common fractions are taught without proof. Only small denominators are used. The study of the metric system is continued. DENMARK.-The seventh school year is the last year of the Folkeskole and the second year of the intermediate or Mellemskole.

In the Folkeskole four hours a week are devoted to mathematics. The study of proportion is continued, and percentage is applied to profit and loss. A comparative study of foreign coins is made.

Geometry. The course in geometry includes the study of triangles, quadrilaterals, and circles. The pupils learn to compute the volume of right prisms and cylinders with given altitudes and bases. Drawings and models of plane and solid figures are made. Instead of making these drawings, girls are taught the method of keeping simple accounts.

Provision is

Intermediate school.-Five hours a week are devoted to mathematics. made for almost daily drill in oral arithmetic. Simple and compound proportion, percentage, simple interest, partnership, simple bookkeeping, and easy mensuration are studied. Geometry is usually taken up before algebra. One or two hours a week are devoted to the study of intuitional or observational geometry. The fundamental properties of straight lines and planes, the measurement of angles, the congruency and similarity of triangles, and the principal properties of quadrilaterals and circles are taught.

ENGLAND. The four fundamental operations for common and decimal fractions are reviewed, and in a few schools some facts relating to recurring decimals are presented. Numerous problems are worked by means of unitary analysis. The pupils receive a good deal of practice in the drawing up of invoices and the making out of bills. The subject of simple interest is taught, most of the problems being worked by means of formulas. The applications of percentage include profit and loss, discount, and taxes. Some easy problems are given in generalized arithmetic. Many rules are expressed with letters. The algebraic equation is introduced, and easy equations are solved. Simple substitution is also taught, and the meaning and use of negative numbers are briefly presented. Pupils are taught to use rulers marked in centimeters and millimeters. Volumes are found by the use of cubic blocks, by graduated vessels, and by emptying or displacement. The rules for the mensuration of various kinds of triangles, of the rhombus and rhomboid are derived. Simple volumes are studied. Squared paper is extensively used for finding the area of irregular figures. The relation between the length of the circumference and the diameter is determined. Pupils graph simple statistics, such as the school attendance, temperature, and prices.

The work in geometry includes the finding of the locus of points equidistant from a given point, the locus of points at a given distance from a given line, and the locus of points equidistant from three given points. The pupils prove in several ways that the sum of the angles of a triangle is 180 degrees. The relation of interior and exterior angles of triangles is discovered, and the method of dividing a line into equal parts or into parts having a given ratio is taught. The equality of triangles is shown by means of superposition. Some attention is given to the study of circles and to inscribed and circumscribed squares.

FINLAND.-The seventh school year is the last year of the primary school. From one to two hours are devoted to arithmetic. The work of previous years is reviewed, and proportion and discount are studied. The study of algebra is begun in the lycée.

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

procedures and abridged computations. In algebra the equation of the first degree is studied. Positive and negative numbers are introduced, and the idea of function is emphasized.

The course in geometry includes the study of triangles, quadrilaterals, and polygons, special attention being given to the study of similarity. The properties of circles are extensively considered, and the pupils are led to discover the rules for areas of plane figures.

HOLLAND.—The seventh school year is the first year of the middle school and of the Gymnasium. The middle school of Amsterdam has a course of five years. Six lessons of 50 minutes each are given weekly in mathematics. The fundamental operations with integers, common and decimal fractions are reviewed. Tests for divisibility, greatest common divisor, and least common multiple are studied, and some attention is given to the subject of circulating decimals. The instruction in algebra is closely related to that in arithmetic. Simple equations in one unknown are studied with short cases of multiplication, and easy types of factoring are presented. Geometry is studied intuitively. The course includes the study of elementary properties of angles formed by parallel and transverse lines; the properties of triangles, quadrilaterals, and polygons.

Gymnasium.-Four hours a week are devoted to the subject of mathematics. Usually one hour is given to arithmetic, two to algebra, and one to geometry. All pupils have the same course in mathematics during the first four years of the Gymnasium. The fundamental operations with integers, common and decimal fractions are reviewed The four operations with algebraic equations are taught. The course in geometry is of a propedeutic nature.

HUNGARY.—The seventh school year is the third year of the Bürgerschule, Gymnasium, and Realschule. In the Bürgerschule six hours a week are usually devoted to mathematics. The course includes the study of simple cases of proportion, easy computations in percentage, simple exercises in physics, and alligation. Some attention is devoted to geometric and free-hand drawing.

Gymnasium and Realschule.-Three hours a week are devoted to the study of mathematics and two hours to geometric drawing. The course in arithmetic includes the applications of percentage, simple and compound proportion, profit and loss, the making and receipting of bills, and the solution of numerous problems based on commercial and national transactions. A good deal of attention is devoted to the subject of savings banks.

The course in geometric drawing includes the construction of plane figures, the reduction and enlarging of figures, and the making of designs.

ITALY. The seventh school year is the third year of the Ginnasio and of the modern school.

One hour a week is usually devoted to the study of practical arithmetic. Geometric drawing and intuitive geometry are taught. The course in algebra includes the formal introduction of literal computation and the study of the four fundamental operations. The course in geometry includes the study of the principal definitions and postulates of formal geometry; the relation of position of perpendiculars, parallels, straight lines, and planes; the construction of trihedral angles, parallelograms, polygons, and the regular polyhedra.

JAPAN.-The seventh school year is the first year of the higher primary and of the middle school.

The subject matter of the instruction in the higher elementary school consists of a more intensive study of the work of the ordinary elementary school, with the addition of ratio and proportion.

In the first year of the higher elementary school fractions, percentage, and ratio and proportion are presented.

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Middle school.-Four hours a week are devoted to the subject of mathematics. The course includes the study of numeration, notation, integers, common and decimal fractions, compound numbers, tests for divisibility, prime numbers, factors, greatest common divisor, least common multiple, and ratio and proportion.

ROUMANIA. The seventh school year is the third year of the Gymnasium. The pupils are taught the tests for divisibility, the greatest common divisor, and least common multiple. Common and decimal fractions are comprehensively reviewed; and the subjects of square root, ratio and proportion, interest, and double entry bookkeeping are studied.

The course in geometry includes the study of the similarity of triangles and polygons, regular polygons, and formulas for the areas of the common plane figures. Geometry is studied intuitively.

RUSSIA. The seventh school year is the fourth year of the Realschule and of the Gymnasium. The course in both types of schools is practically the same. Four hours are devoted to the study of mathematics, two being given to algebra and two to geometry. The pupils study the operations with polynomials, simple fractions, negative numbers, exponents, simple equations with one or more unknown, involution and evolution, and the theory of proportion.

The course in geometry includes the study of the straight line, angles, parallels, triangles, quadrilaterals, polygons, circles, and the fundamental constructions. A large number of numerical exercises are solved.

SWEDEN.-The seventh school year is the fourth year of the Realschule. Five hours a week are devoted to the study of mathematics. Two hours are given to arithmetic, one to algebra, and two to geometry. The instruction is largely of a practical nature. Applications of percentage and simple bookkeeping are taught. The pupils study only enough algebra to enable them to solve simple equations and to factor the simplest types.

The course in geometry includes the simple constructions of triangles and quadrilaterals and the application of geometry to physics. All instruction in geometry is closely correlated with the instruction in drawing.

SWITZERLAND.-The seventh school year is the third year of the middle school and the first year of the Gymnasium.

In the Gymnasium four hours a week are usually devoted to arithmetic and algebra and one to geometry. The work of previous years is reviewed and extended. The study of the operations with common and decimal fractions is completed, and the application of percentage to profit and loss and interest are studied. The course also includes the study of alligation, square root, abridged computations, and foreign money.

The course in geometry includes the use of the ruler and compasses in making simple constructions. The congruency of triangles is established intuitively, and a great deal of attention is devoted to practical measurements and drawings.

UNITED STATES (New York State course).—It is assumed that pupils who have completed the work of the six previous school years should be able (1) to read reasonably large figures at sight and to write numbers rapidly from dictation; (2) to perform all fundamental processes in arithmetic rapidly and accurately; (3) to reason quickly and explain simple problems; (4) to compute with common and decimal fractions without hesitation; (5) to comprehend the fundamental principles of percentage and their application.

During the seventh school year a good deal of attention is given to oral drill in getting approximate results. The daily activities of the pupils are drawn upon freely to make the arithmetic interesting, informational, and practical.

Interest, including the subjects of commercial paper, bank discount, and compound interest, is studied. Tables are used in the computation of compound interest. The pupils are taught how to deposit money and draw checks. Many problems are