lar coordinates. The principal formulas for distances and areas are developed and applied. Since the idea of function has been developed in the previous years, a good deal of time is saved here. The equation of the straight line is presented in four

sections and the tangents for each of the conics are studied.

The conic

The course in algebra includes the study of arithmetical and geometric progression, compound interest, the theory of permutations and combinations, variations, the binomial theorem for positive integral exponents, and the fundamental ideas of the theory of probability.

Realschule and Realgymnasium.—Two hours a week are devoted to the subject of descriptive geometry. Orthogonal and oblique projections are studied, and a good deal of emphasis is put upon perspective drawings. Tangential planes and plane sections are studied. The principal aim of the course in descriptive geometry is to develop spatial concepts and skill in constructions as a basis for the work in more advanced technical institutions.

One hour a week is usually devoted to the subject of geometric drawing.

In the Realschule, more attention is devoted to the study of analytic geometry than in the Gymnasium. The subject is treated as a continuation and extension of the theory of functions, which has been presented in previous years. The straight line is exhaustively studied as a basis for other figures. Practically all computations are made by the use of rectangular coordinates. A good deal of attention is devoted to the study of conic sections. Special attention is given to problems in which conic sections appear as geometric loci. Easy differential and integral calculus are applied to problems in physics. Only those computations are introduced which simplify or make more intensive the knowledge of physics. There is a decided tendency to restrict the theoretic matter to the smallest possible compass.

BELGIUM.—The eleventh school year is the fifth year of the Athénée Royal. In the Greek-Latin and in the commercial Athénée three hours a week are devoted to the study of mathematics. In the Latin and in the scientific sections six hours a week are devoted to the subject.

Greek-Latin section. The algebra and geometry of previous years are thoroughly reviewed. Equations of the first degree with two or more unknowns and simple applied problems are solved. The interpretation of negative, indeterminate, and infinite values is emphasized. Results are generalized whenever possible. The remainder theorem is introduced and algebraic fractions are studied.

The course in geometry includes the study of the properties of circles, chords, secants, and tangents, the congruency and similarity of triangles and polygons, proportional lines and problems in construction. A good deal of attention is devoted to drawing to scale and to out-of-door measurements.

The course in the scientific and in the Latin Athénée is practically the same. The work in arithmetic includes a thorough review of the work of previous years, tests for divisibility, greatest common divisor and lowest common multiple, prime numbers, the theorem of Fermat, common and decimal fractions, approximate computations, complex numbers, and cube root.

The course in algebra includes the solution of equations of the first degree with one or two unknowns, and of simple quadratic equations with one unknown, the square root of binomials, the progressions, logarithms, compound interest and annuities, maxima and minima.

In geometry the study of regular polygons, circles, and transversals is emphasized. The trigonometric functions are defined, and the fundamental formulas are developed and applied. Quite a good deal of attention is devoted to surveying, especially leveling. Numerous figures are drawn to scale.

DENMARK.--The eleventh school year is the second year of the Gymnasium. Two periods a week are devoted to the study of mathematics, in the classical and in the modern language gymnasium, and six periods a week are devoted to the subject in the mathematical-scientific Gymnasuim.

The trigonometric functions are defined and represented graphically, and the principal formulas of plane trigonometry are developed and applied. Some attention is devoted to the subject of goniometry.

The formulas for the surfaces and volumes of regular solids are developed and the fundamental ideas of conic sections are presented. The principal theorems of solid geometry are studied. A good deal of emphasis is put upon the study of spherical triangles and of symmetry and similarity.

ENGLAND. The eleventh school year is the last year of the central school. By the time a boy has completed the work of this year, he should be able to solve any ordinary problem in arithmetic, especially any problem of a practical nature. Workshop methods are emphasized. Constructions and approximations are given a good deal of attention. Many of the problems involve weights, volumes, and density. In some schools simple surveying is given and graphs are taught.

The course in algebra includes the study of quadratics, graphs, fractional coefficients, and indices, formulas, radicals, logarithms, arithmetical and geometric progressions. The geometry of previous years is reviewed and extended, and the work is closely correlated with mechanical drawing and with science. The pupils study the circle, the ellipse, easy vectors, and areas, and make simple constructions.

No

The work in trigonometry includes the study of logarithms and problems involving heights and distances. Angles with given sine, cosine, or tangent are constructed. FINLAND. The eleventh school year is usually the last year of the course. details are given, but when the pupil has completed the course, he has finished the study of algebra up to permutations and combinations. He has studied plane and solid geometry and the elements of trigonometry. In the classical lycees, only the study of the right triangle is included in trigonometry.

FRANCE. The eleventh school year is a one-year course to prepare for the second part of the bachelor degree. It is called the special mathematics course. Eight hours a week are devoted to the study of mathematics.

The theoretical parts of arithmetic are presented. Special emphasis is put upon the theory of prime numbers and of circulating decimals.

The course in algebra includes a review and an extension of the work of the preceding year. Equations of the first and second degrees involving two or more unknowns are solved. Inequalities of the first and second degrees are studied. Arithmetical and geometrical progression, logarithms, compound interest, and annuities are studied. The idea of coordinates is introduced, and some simple equations are ax+b

are

derived. The functions y=ax+b y=ax+b1 y=ax2+bx+c_y=ax*+bx2+c represented graphically. The maximum and minimum values of certain functions having numerical coefficients are found. The areas of curves regarded as functions of the abscissa are computed.

The course in trigonometry includes a review and the application of trigonometry to surveying.

The course in geometry includes a thorough review of the work of preceding years and in addition to this, radical axes, polars, inversions, the theory of vectors, and central projections are studied. The ellipse, hyperbola, and parabola are considered, and their equations are derived. Problems involving tangents to these three curves are solved.

The course in descriptive geometry includes the study of the straight line, plane, circle, cone, cylinder, and sphere. Plane sections of solids are considered. Shadows are studied and the ideas underlying the construction of topographical maps are pre

sented. A good deal of emphasis is put upon the subject of mechanics and cosmography during this year.

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

The course in the gymnasium includes the study of arithmetical and geometrical progression, compound interest, and annuities. Solid geometry in completed and the elements of plane analytic geometry are introduced.

Realgymnasium.-Five hours a week are devoted to the subject of mathematics. Three hours are devoted to algebra and trigonometry, and two to descriptive geometry. The theory of permutations and combinations, the binomial theorem, determinants, and the general theory of equations of higher degree are studied. Plane and spherical trigonometry are completed and applied to the study of mathematical geography.

The course in descriptive geometry includes the graphic determination of points, lines, and planes, and problems involving these. Numerous problems in which the distances from given points, lines, and planes are to be determined are solved.

Oberrealschule.-Two hours a week are devoted to the study of analysis and to the elements of infinitesimal calculus. The algebra of previous years is reviewed. Permutations, combinations, and the binomial theorem are studied. The ideas of limits and of derivatives are introduced. The derivatives for powers, sines, and cosines are formed. The study is closely correlated with physics and mechanics.

Three hours a week are devoted to the study of analytic and synthetic geometry. Coordinate geometry is introduced. The straight line, circle, pencils of rays, tangents, poles, and polars are studied. Some attention is given to the theory of projections. The course in descriptive geometry includes the study of the cylinder, cone, and sphere, and the introduction of perspective drawing.

HOLLAND. The eleventh school year is the fifth year of the middle school and of the Gymnasium. Four hours a week are devoted to the study of mathematics.

Middle school.-The algebra of previous years is reviewed and extended. The study of trigonometry is continued, and circular functions are introduced. Plane geometry is reviewed; and the cone, cylinder, and sphere are studied. The course in descriptive geometry is completed up to the study of curved surfaces. Two hours are devoted to mechanics and one to cosmography.

Gymnasium.-Two hours a week are devoted to algebra and two to geometry. The course in algebra includes the study of quadratics, radicals, fractional and negative exponents, arithmetical and geometric progressions, logarithms and indeterminate equations of the first degree. The study of solids is completed.

In some Gymnasia the pupils who specialize in mathematics and physics take a more advanced course in mathematics. This course includes the above with the addition of maxima and minima, the theory of limits, and some trigonometry.

HUNGARY.-The eleventh school year is the seventh year of the Gymnasium and of the Realschule.

In the Realschule from three to five hours are devoted to the study of mathematics, The course includes the study of arithmetical and geometric progression, circulating decimals, compound interest, and annuities, Government loans, indeterminate equations of the first degree, the binomial theroem, simple formulas involving surfaces and volumes, the study of trigonometry and its applications to surveying and geography.

From two to three periods a week are devoted to the study of descriptive geometry. The course includes orthogonal projections, angles of inclination, the intersection of simple solids by straight lines and planes, shadow constructions, the revolution of plane and simple solid figures about vertical axes, and the introduction of new planes of reference.

Gymnasium.-From two to three hours a week are devoted to the study of mathematics. The course in algebra includes the study of compound interest and annuities,

Government loans, the progressions, circulating decimals, and the theory of quadratics.

The course in geometry includes the study of the coordinates of a point, the distance between two points, the graphs, and the important theorems of solid geometry, especially those involving surfaces and volumes.

ITALY.--The eleventh school year is the second year of the licée. Six hours a week are devoted to the study of mathematics. Books IV, V, VI, XI, and XII of plane and solid geometry are completed. The theory of proportion, power, equations of the first and second degrees with one and more unknowns, radicals, and progressions are studied. The elements of trigonometry are introduced.

JAPAN. The eleventh school year is the fifth and, in some schools, the last year of the middle school.

Four hours a week are devoted to the study of mathematics, two hours being devoted to geometry and two to trigonometry.

The course in geometry includes the applications of proportion, the study of areas and loci, straight lines and planes, solid angles, polyhedra, the prism, the pyramid, the sphere, the cylinder, and the cone.

The course in trigonometry includes the functions and graphic representation of the functions for acute and obtuse angles, the development and use of the principal formulas, the solution of right and oblique triangles, the use of logarithmic tables, and the applications of trigonometry to surveying.

ROUMANIA.-The eleventh school year is the third year of the lycée. Four hours a week are devoted to the study of mathematics. A good deal of emphasis is put upon mechanics.

The course in algebra includes the study of sines, convergence, the number e, exponential functions, logarithms, derivatives of exponential, logarithmic and circular functions, homogeneous functions, and complex numbers.

In descriptive geometry, estimates of projections, the representation of a point by the aid of projections of two perpendicular planes, and the projection of a straight line are considered.

The pupil's knowledge of algebra and of geometry is utilized to a great extent in the study of mechanics.

RUSSIA. The eleventh school year is the eighth and last year of the Gymnasium. From three to four hours a week are devoted to the study of mathematics. A complete review of the previous year's work in mathematics is given, one hour a week being devoted to the review in arithmetic. The factor theorem and the transformation of equations with the unknowns in the denominator are presented..

SWEDEN.-The eleventh school year is the third year of the Gymnasium. In the Latin Gymnasium, three hours, and in the Realgymnasium, four hours are devoted to the study of mathematics.

In the Gymnasium, trigonometry and solid geometry are completed. Arithmetical and geometrical progresssion and compound interest are studied. Graphs of simple functions, such as y sin x, are introduced.

In the Realgymnasium, in addition to the above course, analytic geometry is introduced and a good deal of attention is devoted to linear drawing. A thorough study is made of the theory of shadows.

SWITZERLAND.-The eleventh school year is the fifth year of the Gymnasium and the third year of the Realschule. In the literary Gymnasium two hours a week are devoted to algebra and one and a half hours to geometry. In the Realgymnasium two hours are devoted to algebra, one to descriptive geometry, and one to bookkeeping. In the Realschule two and a half hours to algebra, three to geometry, and three and a half to descriptive geometry.

Literary Gymnasium.-Logarithms, exponential equations, the progressions, compound interest, annuities, indeterminate equations of the first degree, and plane trigonometry are studied.

Realgymnasium.-In addition to the above course, continued fractions, equations of the second degree in two unknowns, and solid geometry are studied. Goniometry and trigonometry with its applications are introduced.

Realschule. The course in the Realschule is the same as in the Realgymnasium, with the addition of the approximate solutions of equations of higher degree. The elements of spherical trigonometry applied to geography and the elements of plane analytic geometry are also introduced.

In both the Realgymnasium and Realschule descriptive geometry is studied. The course is more intensive in the Realschule. The relation of points, straight lines, planes and simple solids in vertical and horizontal projections are considered. The fundamental problems in construction are presented. The projections of the circle are studied, and triangles, prisms, pyramids, and regular solids are given special attention.

UNITED STATES.-The eleventh school year is the third year of the secondary school. The courses in mathematics vary somewhat, but in most of the schools a second course in algebra is given during the first semester, and solid geometry is studied during the second semester.

The course in algebra includes the study of radicals and exponents, quadratic equations with one and two unknowns, the theory of quadratic equations, equations in the quadratic form, the progressions, and the binomial theorem.

The course in solid geometry includes the study of Books VI, VII, VIII, and IX.

SUMMARY OF THE ELEVENTH YEAR'S WORK.

The eleventh school year is the third year of the secondary school of the United States. In most of the schools the course in mathematics includes a half year of algebra and a half year of solid geometry. In a few of the schools, the first half year is devoted to the study of solid geometry and the last half to the study of trigonometry.

In the schools of Europe the work in algebra during the eleventh school year includes the study of arithmetical and geometrical progressions, permutations and combinations, the theory of probability, the binomial theorem, and determinates. The preceding topics are included in the most advanced courses in Austria, France, Germany, Holland, Hungary, Sweden, Switzerland, and Roumania. The subject of trigonometry, which was completed during the tenth school year, is applied to surveying. Much more emphasis is put upon the subjects of cosmography and mechanics than in the United States. Descriptive geometry is studied in Austria, Germany, France, Holland, Hungary, and Switzerland, and the study of geometric drawing is continued in Austria. Analytic geometry is introduced in certain of the schools of Austria, France, Germany, Hungary, and Sweden, and the elements of differential calculus are introduced in a few of the Realschulen of Austria and in the Oberrealschulen of Germany. The calculus is applied in the study of physics. Spherical trigonometry is taught in several of the countries, and it is usually applied in the study of mathematical geography.

When a European boy has completed the eleventh school year, if he has elected the scientific course, he has studied more mathematics than is offered in any except a very few of the most progressive secondary schools in the United States.