MATHEMATICS AT WEST POINT AND ANNAPOLIS GENERAL REPORT Of all the technical schools in the United States, probably none exists whose aim is so clearly defined as that of our two great Government schools for the training of officers for the United States Army and Navy. The purpose of these schools is strictly utilitarian, viz, to give to a selected number of young men of this country the best possible technical training for the positions of responsibility in the Army and Navy. That these schools do their work well is amply demonstrated by the efficiency of our military and naval forces. It is always of interest to see how a school fulfills its aim, how it adjusts its subject matter to meet the particular task in hand, and how it eliminates that which detracts from the main purpose. To teachers of mathematics such an investigation will be of particular interest, especially in view of the tendency of a number of modern educators to demand a justification of every topic in the curriculum. It is the purpose of the two reports submitted herewith to give such an exposition of the teaching of mathematics in these two great schools, the United States Military Academy at West Point and the United States Naval Academy at Annapolis, these being the only schools of the kind in the United States. These reports will repay a careful reading; they are suggestive to all teachers not only in details of class organization, but in the general handling of subject matter to serve a definite purpose. They will also show why mathematics has so long held and still retains its prominent place in the training of military and naval officers. To those who are familiar with similar schools abroad, it will be of great interest to compare the aims and results of instruction in mathematics in a famous Government school like the École Polytechnique1 of Paris with those found in West Point and Annapolis. In the French school the greatest prominence is given to purely theoretical instruction in higher mathematics, with a very limited amount of practical application; in our two schools few topics are taught beyond the essentials of the calculus, and the practical problem is the basis and end of all the work. 1 The work of the École Polytechnique is described in the report of the French Committee (published by Hachette, Paris) of the International Commission on the Teaching of Mathematics. AIMBOTLIAD REPORT OF SUBCOMMITTEE 1 MATHEMATICS IN THE TRAINING OF ARMY OFFICERS, INCLUDING SCHOOLS FOR GRADUATES OF WEST POINT The training of Army officers is carried on in the following schools: (1) The Undergraduate school at West Point; (2) the Engineer School at Washington Barracks; (3) the Ordnance School at Sandy Hook, N. J.; and (4) the Coast Artillery School at Fort Monroe. I. THE UNITED STATES MILITARY ACADEMY HISTORICAL SKETCH The United States Military Academy, established in 1802, is situated at West Point, N. Y. It is a school for the practical and theoretical training of cadets for the military service. The students of the academy are known as "cadets." Upon completing its course satisfactorily cadets are eligible for promotion and commission as second lieutenants in any branch of the military service of the United States. The Corps of Cadets as now constituted consists of 1 from each congressional district, 1 from each Territory, 1 from the District of Columbia, 1 from Porto Rico, 2 from each State at large, and 40 from the United States at large, all to be appointed by the President. Those cadets appointed from States or Territories must be actual residents of the congressional or Territorial districts, or of the District of Columbia, or of the States, respectively, from which they are appointed. Four Filipinos, one for each class, are authorized to receive instruction as cadets, to be eligible on graduation to commissions only in the Philippine Scouts. Under the apportionment of Members of Congress, according to the Twelfth Census, the maximum number of cadets is 533. The total number of graduates from 1802 to 1909, inclusive, is 4,852. ENTRANCE REQUIREMENTS The United States Military Academy is the only school in the country that has for its sole object the furnishing of commissioned officers to the United States Army. From the method of selection of appointees referred to above, it is reasonable to believe that the Military Academy receives a more varied class of students and one more broadly representative of all the States than any other educational institution in the country, except the similarly constituted Naval Academy at Annapolis. Entrance to the academy is by examination. The scope of the entrance requirements in mathematics is as follows: Algebra.-Candidates will be required to pass a satisfactory examination in that portion of algebra which includes the following range of subjects: Definitions and notation; the fundamental laws; the fundamental operations; factoring; highest common factor; lowest common multiple; fractions, simple and complex; simple or linear equations with one unknown quantity; simultaneous simple or linear equations with two or more unknown quantities; involution, including the formation of the squares and cubes of polynomials; binomial theorem with positive integral exponents; evolution, including the extraction of the square and cube roots of polynomials and of numbers; theory of exponents; radicals, including reduction and fundamental operations, rationalization, and equations involving radicals; operations with imaginary numbers; quadratic equations; equations of quadratic form; simultaneous quadratic equations; ratio and proportion; arithmetic and geometric progressions. Candidates will be required to solve problems involving any of the principles or methods contained in the foregoing subjects. Plane geometry.-Candidates will be required to give accurate definitions of the terms used in plane geometry, to demonstrate any proposition of plane geometry as given in the ordinary textbooks, and to solve simple geometric problems either by a construction or by an application of algebra. These entrance examination papers are prepared at the academy. They are furnished to examining boards of Army officers convened annually at such places as the War Department may direct. The examinations are thus held at points widely distributed over the United States and its dependencies. All papers are sent to the Military Academy for correction. DISTRIBUTION OF TIME A cadet when admitted to the Military Academy must be over 17 and under 22 years of age. He pursues a course of study lasting 4 years and 3 months. The instruction in pure mathematics extends from entrance on March 1 to March 1 two years later. This time is subdivided as follows: Each period is 1 hour and 20 minutes long. The instruction in applied mathematics is distributed through the last three years of the undergraduate curriculum, except that surveying follows immediately upon the completion of trigonometry. The time assigned to each subject is as follows: EXTENT OF COURSE IN PURE MATHEMATICS The course in pure mathematics as laid down in general terms in the regulations of the academy is: Geometry. Problems in plane geometry. Geometry in space; planes, lines, polyhedrons, cylinders, cones, and the sphere. Algebra. Detached coefficients, factoring, linear and quadratic equations, graphic representation, the binomial theorem, inequalities, theory of exponents, complex numbers, ratio and proportion, variation, progressions, series, undetermined coefficients, logarithms, the slide rule, interest, combinations, probabilities, determinants, and the theory of equations. Trigonometry. The measurement of angles, the simple trigonometric functions, the functions of several angles, trigonometric equations, and the solution of plane and spherical triangles. Analytic geometry. In the plane: Systems of coordinates, change of axes, anharmonic ratios, the straight line, the circle, the parabola, the ellipse, the hyperbola, the polar equation of the conic, the general equation of the second degree, systems of conics, and envelopes. In space: Systems of coordinates, transformations of coordinates, the plane, the straight line, surfaces of the second degree, and plane sections. Descriptive geometry. Orthographic projections: Points, right lines and planes, problems on their relative positions, the revolution of objects, curves in space (especially the circle and cylindrical helix), surfaces (including ruled surfaces and surfaces of revolution), tangent planes, intersection of surfaces, trihedrals. Spherical projections. Shades and shadows. Central projections or perspective. Isometric projections. Differential and integral calculus. The theory of limits, differentiation of the standard elementary forms, simple applications of the derivative to velocity and rates, maxima and minima, curvature, evaluation of indeterminate forms. envelopes, expansion of functions, curve tracing, tangent planes, and normal lines to surface. The fundamental principles of integration, the integration of the standard forms, the measurement of arcs, areas, and volumes, mean values, approximate integration, ordinary differential equations. Method of least squares. Errors to which observations are liable, the correction of observations, the probability curve and its equation, measures of precision, the deduction and the application of the formulas for probable and mean error, weights of observations, and the formation of equations of condition and normal equations. The course of mathematics as outlined above is compulsory for all students and the time devoted to the subject is the same for all. A marked difference, however, is made in the amount and the difficulty of the matter required of students differing in ability. This is accomplished in the following way: Upon admission the students are arranged in sections of 10 to 12 in order of ability as shown by their marks on their entrance examination in mathematics. This is a tentative grading of no great value, and in consequence all sections have identical courses until the examination which takes place at the end of the first three months. At that time a welldefined grading according to mathematical ability is possible. Each class is then separated into three subdivisions, known as upper, middle, and lower thirds. A standard course in mathematics has been set for the lower third that is believed to contain the minimum of knowledge, a proficiency in which will insure that the student can intelligently pursue the applied courses that follow. A more extended course is provided for the middle third. A still more comprehensive course is provided for the upper third, one that is expected to give a thorough foundation. in pure mathematics for all subsequent undergraduate and graduate work. The difference in the courses for the groups of a class appears in the use of a more comprehensive or difficult textbook for the abler student, or, if the same text be used, in an omission of the more difficult subjects for the less able; also in the grading of the illustrative problems and exercises assigned to the different groups; also in a further advance into the theory of the subject by the abler student, even to the inclusion of an additional branch of mathematics. Transfers of students are freely made from time to time between the sections in each group and less frequently from one group to another. These transfers are based upon the quality of the work performed by the student as proved by his daily tasks, oral and written. Each group is thus induced to strive for the full development of its mathematical powers. A daily alternation throughout the course, where possible, of two subjects-usually one geometric and the other analytic-and toward the end of the mathematical course a similar alternation of mathematics with the course in mechanics enable the various branches to supplement and assist each other and also give the student more time for the digestion of each subject or for concentration upon the one that demands of him the greater labor. In the fifth class algebra alternates with plane and solid geometry. In the fourth class algebra alternates with plane and spherical trigonometry. Later, analytic geometry alternates with descriptive geom |