SUMMARY OF THE TWELFTH YEAR'S WORK.

The twelfth school year is the last year of the secondary school in the United States. In many of the schools the first half of the year is devoted to the study of solid geometry and the last half to trigonometry or to business arithmetic. In some of the schools the first half of the year is devoted to the study of plane trigonometry and the last half to the study of college algebra. It is rare that courses in spherical trigonometry, beyond the study of the right spherical triangle, are offered in the United States.

In practically all of the European countries the twelfth school year begins or ends with a comprehensive review of the mathematics of the preceding years. Special emphasis is put upon such reviews in Austria, Germany, Denmark, Holland, and Hungary. The practical applications of mathematics are emphasized in most of the countries, and in Austria and Germany some attention is devoted to the history of mathematics. Spherical trigonometry is offered in the schools of Holland, Germany, Hungary, and Switzerland; and analytic geometry is studied in Denmark, Germany, Austria, Hungary, Sweden, Switzerland, and Roumania. In a few of the Cantons of Switzerland solid analytic geometry is studied. Descriptive geometry is offered in Germany, Austria, Hungary, and Switzerland, and advanced algebra is studied in most of the countries. Differential and integral calculus are offered in the schools of Austria, Belgium, Denmark, France, Sweden, Switzerland, Russia, Germany, and Roumania.

The relations between algebra and geometry are especially emphasized in France, and the relations between mathematics and physics receive special emphasis in Germany, Holland, and Switzerland.

When a European boy has completed his twelfth school year he has had the opportunity of studying more mathematics than is offered in any of the secondary schools of the United States. He has had more practice in applying his mathematics in physics, cosmography, and mathematical geography than is the case with the American boy. The simultaneous study of several mathematical subjects results in a more complete mastery of each. He sees the unity of mathematics in a way that is seldom true with the American boy. He can use his arithmetic and algebra in the solution of geometrical problems and his arithmetic and geometry in the solution of algebraic problems much better than the average American boy. He has some knowledge of analytic geometry and of the infinitesimal calculus. The frequent drills and reviews so common in European schools have furnished him with a large number of mathematical facts and formulas that he can use more readily than his American brother. Mathematics to him is an interesting and a fruitful subject, because he has learned to appreciate something of its deeper significance.

XIV. THE WORK IN MATHEMATICS IN THE THIRTEENTH SCHOOL YEAR.

In Belgium and in Switzerland the course in the secondary schools extends over a part of the thirteenth school year.

In Belgium the thirteenth school year is the seventh and last year of the Athénée Royal. In the commercial course two hours a week, in the Greek-Latin three hours, and in the Latin and in the scientific courses eight hours a week, are devoted to the study of mathematics.

Greek-Latin course.-The subjects of geometry and algebra are thoroughly reviewed, and the progressions, logarithms, annuities, and loans are especially emphasized. The theorems for the surface and volume of the prism, pyramid, cone, and sphere are applied in numerous problems. Spherical geometry is given a good deal of attention.

The fundamental formulas of trigonometry are studied and applied in the solution of triangles. The application of trigonometry to surveying is emphasized, and the pupils learn to use the surveyor's instruments.

Scientific and Latin courses.-Two hours a week are devoted to review and to new applications of the mathematics of previous years. Determinants are studied and applied in the solution of equations. The principal theorems of spherical trigonometry are developed and applied. The study of analytic geometry is continued and the principal formulas for the straight line and circle are developed and applied. Both rectilinear and polar coordinates are used, and the tables are taught for the transformation of coordinates. Poles and polars are extensively studied, and the equations of conic sections are developed.

The course in descriptive geometry includes the study of the principal theorems relating to points, straight lines, and planes; simple rotations are introduced.

In most of the Cantons of Switzerland the final year of the course lasts but one semester. A thorough review of the mathematics of previous years is given, and the study of plane analytic geometry and calculus is extended.

Tables 2 to 6 indicate the time at which each subject is introduced into the various types of schools and the school years during which the subject is taught.

The nomenclature of some of the mathematical subjects varies to such an extent in the various countries that it is not possible to be exact in all details in such tables. These tables are intended to indicate usual practices, and not exceptional cases.

The term geometry, as used in Table 4, includes all that is usually thought of in this 'country under the headings of observational, demonstrative (plane and solid), and descriptive geometry and geometric drawing. In many cases the reports of the countries do not indicate the divisions between two types of geometry sufficiently to justify the construction of a separate graph for each.