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etry. In the third class the course in the calculus and least squares alternates during its latter portion with the subject of mechanics.
PURPOSE OF TEACHING MATHEMATICS
The extent of the mathematical curriculum is determined primarily by the requirements of the succeeding scientific courses in this academy and at the graduate schools for engineers, ordnance and artillery officers.
Remarks on the purpose of the study of mathematics have been variously given in the following extracts:
(From report of Congressional Committee on Military Affairs, 1834.) Mathematics is the study which forms the foundation of the course. This is necessary, both to impart to the mind that combined strength and versatility, the peculiar vigor and rapidity of comparison necessary for military action, and to pave the way for progress in the higher military sciences. All experience shows that the mind, in order that it may act with efficiency, must be accustomed to exertion. It should be taught gradually to develop its own powers, and as it slowly learns their capacity and the manner of employing them, the increasing lights which are thrown upon its course will enable it to go on for an unlimited extent in the path of improvement.
(From a report of the Academic Board of the Academy, 1843.)
The academic board believe that one of the most important objects of the academy is to subject each cadet, previous to his promotion to a higher grade in the Army, to a thorough course of mental as well as military discipline, to teach him to reason accurately, and readily to apply right principles to cases of daily occurrence in the life of the soldier. They are satisfied that a strict course of mathematical and philosophical study, with applications to the various branches of military science, is by far the best calculated to bring about this end, and that the present scientific course at the academy, the result of the experience of many years, is in its main features such a course. They are aware that many of the cadets, as is the case with most of those who pursue a scientific course at other institutions, will have little occasion to make practical applications of the many mathematical formulæ with which they meet, and that they may have passed over particular problems without thoroughly understanding their meaning in all their important points; still, if the course has been carefully taught, the reasoning faculties will have been strongly exercised and disciplined, and a system and habit of thought acquired which are invaluable in the pursuit of any profession, and as desirable for the infantry or dragoon officer as for any other officer in service.
(From a report of Prof. Church, 1860.)
I consider the course of mathematics as now taught to all who pass their final examination sufficient to enable the cadet to acquire a thorough knowledge of all the courses which follow it, and not more than sufficient to enable him to study with advantage the courses of natural and experimental philosophy, engineering, and ordnance and science of gunnery. Moreover, I regard the mental training of the pupil as one of the great objects of the study of mathematics, a training particularly required by the officer of every corps of the Army, and to which many of them owe their distinguished success; and I believe that the scientific reputation of the academy depends in a great degree
upon the thoroughness and extent to which the mathematics and its applications to other sciences are taught, and to diminish them would seriously affect this reputation and the success of the institution.
(From a report of Prof. Bass, 1896.)
The object of the mathematical instruction in this academy is primarily to prepare the pupil for the study of mechanics, wave motion, astronomy, electricity, ordnance and gunnery, and engineering. In addition the study of mathematics develops the reasoning faculties and establishes a mental foundation upon which any branch of knowledge may safely and rapidly be constructed.
(From a report of Prof. Echols, 1907.)
It has been thought best in the past to educate all of our officers in the same institution with practically the same curriculum, thus giving to all branches of the service a nucleus of men well trained along scientific lines and qualified to take up intelligently any graduate work that may be demanded of them ***.
If we choose to draw conclusion from the xample of other countries, it is that less mathematics is taught on the average to officers of infantry and cavalry. On the other hand, the strides in science in recent years and the increased application of mathematical investigation in electricity, chemistry, ordnance, and engineering demand a thorough foundation in scientific training for officers of engineers, ordnance, and artillery. This foundation must be laid upon a good and sufficient training in pure mathematics. I find that in range the course in mathematics pursued at our Military Academy by the upper sections compares favorably with theirs, constituting about an average. The French and Italian cadet schools cover more ground, the English, Austrian, and German slightly less. In England and Germany there are, however, courses of pure and applied mathematics for specially selected officers at graduate schools.
The conclusion seems to be that the range covered by the upper course in mathematics at West Point is not too great for the proper training of officers of engineers, ordnance, and artillery; that its value may be enhanced by a more thorough training in the subject of differential equations at the expense, if need be, of some other subject.
Since this Government is disposed to take due pains with the education of its officers, irrespective of branch, and since cadets at our institution do not before admission cover with certainty as much ground as at the preparatory schools of other countries, a general course in mathematics should be required of all officers for its practical value, but no less for its educational value in training the mind to logical forms of thought, in developing the sense of absolute truthfulness, together with a confidence in the accomplishment of definite results by definite means. A special course in mathematics is, on the other hand, to be regarded as the foundation stone in the training of officers for the scientific corps.
For the purpose of control and instruction, the personnel of the department of mathematics consists of a professor, an associate professor, an assistant professor, and a number of instructors, varying with the size of the classes under instruction. This staff is composed entirely of officers of the United States Army. The professor is permanently in charge as the executive head of the department. He
has military control over the instructors and students while they are on duty under him; he assigns the instructors to their work, makes out the courses of study, superintends the instruction, and is responsible that approved methods are followed.
His juniors are all younger officers detached by the War Department from the several branches of the service and sent for periods of four years or more to serve on the teaching force of the academy. They are nominated to the War Department by the professor. Their selection is dictated by the fact that they have shown special ability, aptitude, and predilection for mathematics by their undergraduate record; also by the fact that they are believed to possess those other important qualities of the instructor that are essential for the molding of character in the student.
The junior instructors, whenever it is possible, are trained upon the course they are assigned to teach by a series of conferences conducted by the professor or one of the more experienced instructors.
Under normal conditions only one-fourth of the body of instructors is changed annually. This insures continuity in the teaching staff.
An instructor has charge of two sections of about 10 cadets each which recite to him daily (6 days per week).
METHODS. REVIEWS, AND EXAMINATIONS
A certain portion of a standard textbook is assigned in advance for each day's recitation. Upon this theory and upon the exercises applying it the student is expected to spend from three to four hours. in preparation. When the section reports to the instructor for recitation, the latter devotes as much of the beginning of the period as he judges best in answering the questions of the student upon the different points of the lesson for the day, solving illustrative problems and volunteering such explanations and elucidations of the subject matter as he may think advisable. An endeavor is made to select textbooks that present the subject matter in sufficient detail to enable an earnest student of normal intelligence by thorough application to comprehend clearly the theory enunciated. It is the purpose to require this thoughtful application which develops the spirit of mental independence and consciousness of power. A lack of proper effort to accomplish his daily task, therefore, is regarded as a military delinquency on the part of the student and is punished as such.
It is the custom, though not an invariable one, to take each subject of study three times.
First, an advance is made, during which the lessons are short and explanations and demonstrations are numerous. The student is expected to ask and to receive assistance from his instructor upon any
difficult point. He is questioned and induced by hints from the instructor to originate questions upon matters that may have failed to suggest themselves to him. On the advance all work of instructor and student is performed upon the blackboard to the benefit of the entire section. It is believed that much profit is derived from seeing the errors of a fellow-student's work corrected and also from observing its excellencies. The sections are composed of pupils of approximately equal ability. Each member is thus encouraged to take an active part in any discussions and receives his full share of the instructor's time and attention.
At regular intervals a review of the advance work is made. The lessons are longer, fewer explanations are necessary, and more applications of theory to the solutions of problems are required of the student. Occasionally an entire period on this partial review is devoted to the solution in writing of the same set of problems by all cadets of the same group.
A second or general review takes place at the close of each subject. This covers the entire course, in long lessons continued over several days. Each day a thorough written test is held upon the portion of the subject assigned for that day. This includes both demonstrations and practical applications. After the completion of the work by the student, solutions are given and explained in full at the blackboard by an instructor and questions relating thereto are answered by him. The student who has qualified upon these written tests is considered proficient on the work of the term without further examination. The one who has not so qualified is given a regular written examination of four to eight hours covering the entire
COURSE IN MECHANICS AND ASTRONOMY
The courses in the calculus and in mechanics begin at the same time; the calculus is taught daily from September 1 to November 1, and alternates daily with mechanics from November 1 to March 1; mechanics is taught on alternate days from September 1 to June 1.
The lessons in mechanics in September and October are two-hour periods in the section room, without a preparatory study hour. The principles developed in this part of the course relate to so much of the subject of statics as does not involve the use of the calculus. Simple illustrative experiments are made by the pupils, and the first part of the text is followed in such a way as to bring out the essential points with the least possible loss by misdirected study. The purpose of this part of the course is not only to master the fundamental principles of statics, but also to direct the study so that the pupil shall always use his mathematical knowledge and the language of the text merely as tools with which to shape accurate physical con
ceptions of the subject. During this preliminary part of the course only such simple definitions as are necessary to the work in hand are required of the students, and the experience gained here is an effective preparation for mastering the more difficult definitions and distinctions in the subject of kinetics.
By November 1 the class has covered enough of the calculus to take up the study of mechanics with mutual advantage to the two courses, and the recitations in mechanics from that date have a study period for preparation, alternating with the calculus.
It is the aim of the course to cover the subject as thoroughly as practicable in a little more than 100 lessons; or, stated in the proportion of the student's entire work for an academic year, about one-fourth.
The subject of statics is carried far enough to include the mechanical powers and a short treatment of the simple principles of graphic statics. The course in rigid kinetics is carried up to as general an explanation of simple gyroscopic motion as may be made by differential equations, without the time integrals; and the year's work is concluded by a short study of fluid equilibrium and flow. Astronomy is carried far enough to secure some facility in handling the sextant, and in solving time, latitude, and longitude problems. This work occupies about 54 days.
The course in ordnance and gunnery requiring previous mathematical training is, in general terms, as follows: Interior ballistics; calculations of the effects of explosion; calculations of the strength of guns of both built-up and wire-wound construction; calculations relating to the construction of the rifling curves of guns; recoil and recoil brakes; exterior ballistics; calculations of the strength of projectiles and the arming resistance of fuses; calculations of the forces brought on the parts of gun carriages by the discharge of the gun; calculations of the stresses in gun carriage parts; calculations relating to gears and gearing; and calculations relating to recoil springs.
II. ENGINEER SCHOOL OF APPLICATION
WASHINGTON BARRACKS, D. C.
OBJECT OF SCHOOL
This school aims to give to the junior officers of the Corps of Engineers a theoretical and practical knowledge in the various departments of engineering required of them in their professional duties as engineers in charge of fortifications, rivers and harbors, lighthouses and aids to navigation, and military engineering in time of peace and war.