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required for admission or as required in the course. These subjects are: arithmetic, algebra through quadratics, plane and solid geometry generally required for admission, and plane and spherical trigonometry, analytical geometry and calculus taken in the course. Descriptive geometry is so interwoven with drawing and other work that its consideration may be omitted. Analytical mechanics has been considered as applied mathematics, and is also omitted.
A glance at this table shows that four do not require an examination in arithmetic. Why this is omitted is not clear. A thorough examination in arithmetic should be given, for it is perhaps the most important branch to the engineer, both on account of its every-day usefulness and by reason of the bearing of its principles on later subjects. The writer's experience is that its teaching is not well done, and that the average student is lamentably deficient. I may add that the old-time mental arithmetic is very valuable for every engineer.
In all these schools algebra through quadratics is required for admission, and one adds that advanced algebra will be required after 1894. All but one require plane geometry, but seven do not require solid geometry. Two of these will after this year. One school requires plane trigonometry, and another will require plane and solid trigonometry and advanced algebra after this year. The condition of high schools throughout the country makes it both feasible and desirable to include all of geometry in the admission requirements. Whether it is best to include trigonometry, unless the school is fed principally from a preparatory school of its own, is not so clear. The drill obtained in classes in high schools, where so few will appreciate the value of trigonometry, is likely to be below the necessary standard, and such a course should be supplemented in the engineering school with a thorough review and a drill in computations and use of tables. The main object, of course, in requiring this subject for admission is to permit the completion of mathematics early in the course, and thus allow the allied subjects to be considered in the earlier years:
Two schools do not require spherical trigonometry, a commendable omission, perhaps, if sufficient drill in plane trigonometry be given.
In addition to the subjects mentioned for the regular curriculum, one school includes differential equations and the method of least squares. The former is of especial value to students in advanced physics, but it is doubtful whether the average engineering student will not be sufficiently prepared without it. The latter is usually given in some form or other in geodesy or other subjects involving the value of observations and averages.
The amount of time devoted to pure mathematics by these schools is nearly the same.
With the exception of two schools, the difference between the mean, 13.7 per cent. of the entire course, and 10.5 per cent. on one side and 15.7 per cent. on the other, is not great, when the possible differences in method of counting are considered. Even in the time devoted to calculus, there is substantial unanimity. The consensus of opinion seems to be that arithmetic, algebra, geometry, trigonometry, analytical geometry, and calculus are essential to the economical training of the engineering student, and that 11 to 14 per cent. of the student's time should be given to these subjects. This unanimity in time and subject matter may not mean a uniformity in methods and thoroughness of teaching.
Instructors generally will agree that all these subjects are economically necessary in the instruction of the engineering student and that the time spent in this training is not extravagant. It would even seem out of place to enter a discussion on this topic. Engineers, however, sometimes object to the calculus, and statements are made that this branch is never used in practice. Even if it had no value to the practicing engineer, its economick value in the presentation of other subjects would cause its retention. A value of mathematical training lies in the gain in quickness and clearness of conceptions, in briefness and accuracy of language in description and explanation. The gain in time and comprehension in other subjects is little understood by these critics. Who would attempt to teach resistance of materials and thermodynanics without it? He would find it an economy of time to teach the calculus first. Besides, calculus is something more than differentiation and integration. The conceptions of rates of change, of the relation of the change of one variable to that of another, of maxima and minima, and of summation are so fundamental to most engineering investigations that it would be folly to. attempt such a consideration without a thorough training in this line. Engineers do use the methods and conceptions of the calculus, generally in an indirect way, although they may not realize it, and, thus they fail to appreciate its educational value.
However, there are criticisms made on the results obtained in mathematical training, criticisms that are heard from engineers and instructors. Students do not have that command of mathematical methods, nor that comprehension of the processes, nor that working knowledge of the details, which may be rightfully expected and which is necessary for its use as a convenient tool. The discipline acquired is far less than it is generally asserted to be. The aim of the student is too often the acquirement of only enough of the subject to pass the examination.
Where then, lies the difficulty, if there be one? It does not lie in the selection of the branches taught, nor in the time given to their treatment. The writer is quite reluctant to offer any suggestion for improvement, since the circumstances surrounding the teaching of mathematics are so diverse, but he will suggest a few items which may or may not be considered applicable to the present condition of mathematical teaching.
"Not how much, but how well" should be the precept continually before the instructor. Teach the elements well. Omit much of the complex portions of the subject, or leave it for an advanced course. Don't attempt to cover too much ground. The new concepts require time for comprehension, and the student who is well drilled in the elements will readily take up more complex problems when necessity requires. It is far better for him to master the rudiments and use them as readily as he uses the operations of multiplication and division than to follow through long demonstrations whose meaning he barely comprehends and whose processes are repeated probably through a feat of memory. In general, then, cut out all parts which may be omitted without seriously affecting the succeeding subjects. Then drill the students thoroughly, giving as much individual instruction as possible. It is only persistent, repeated drilling and practice that gives the average student a fair working knowledge of mathematics.
Illustration and application of the principles should be made in every possible way. A dry principle or an abstract statement must have life and reality given it. The bare demonstration may be meaningless without an appreciation of its use. Something tangible must be given before a new concept is comprehended. Success in this direction depends upon the personality of the instructor.
Avoid confining the training to exact and purely theoretical conditions. Let the student see that he may have very indefinite data. Require him to discriminate in the choice of data. Give problems rather than examples—and engineering problems as far as possible. Correct these and grade them, so the student may know how bad his mistakes are and where his methods may be improved. There is much drudgery in