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way. It is quite within the possibilities of a course of study in engineering to convey to the student healthy and well-balanced ideas as to the influences which will affect the cost of his work, and as to the elements on which the complete cost is to be based, and no school of engineering should be considered as satisfactorily discharging its responsibilities unless it accomplishes that end.

The efficient organization of engineering offices and that of forces and plants for the execution of specific engineering works, as well as the administration of large interests involving engineering operations, such as railways and public works, are all matters which should, if possible, be presented to the student, but in ways that are appropriate to their various characters and consistent with the amount of time and effort which must be bestowed on those other subjects which are fundamentally essential to the course of study. While it is desirable that the student should be brought in touch with these matters, so that he may know that they exist, and, at least approximately, what they are, they are spoken of here thus guardedly because they are not absolutely essential. He can, at best, but derive general impressions at long range, which will have to be materially corrected and developed in their full detail in his practical career; and it is neither necessary nor wise that any considerable portion of his time should be devoted to these administrative features of his later practice, which do not directly affect his immediate professional duties. These observations are not intended to be applicable to post-graduate studies in engineering, which constitute a field of work for the student in which an extended study of organization and administration can most profitably be made.

The method of instruction to be pursued in the school which affords the ideal education in engineering, must be of such a character as to yield the best results with the least amount of unproductive labor to the student, and at the same time train him in the ways of vigorous and independent thought. Dry and lifeless text book recitation work yields by itself little enough that is desirable, while the pure lecture method is equally unsatisfactory. The one is uninteresting routine and utterly ill adapted to the development of mental fertility and strength, while the other involves a mass of misdirected labor on the part of the lecturer, which, without supplementary exercises, leaves the student to shift for himself, without producing much of any result. The main purpose is to convey instruction, and it should be done in such a way as to induce the student to do his own thinking. No instructor can justify himself in merely puzzling the student with grotesque problems, or in harrassing him with abnormal difficulties, but on the other hand the student should be made constantly to feel himself a working part of the system, and that he cannot, without serious loss to himself, take a mere passive part. Unsupplemented lecturing on engineering subjects will not do this, although a very considerable amount of lecture work must be done. The fundamental groundwork of principles with the attendant analytical work and the usual or common applica

tions can most advantageously be put into the hands of the student, either in the shape of a reference or text book or prepared notes, so that the ground to be covered by the lecturer may be carefully worked over by him prior to the lecture hour. The lecturer then need elaborate only the obscure or difficult portions of the subject matter and the applications to living engineering questions in this manner. The student will be induced by reasonable and natural methods to acquire a degree of self-reliance and strength in approaching new problems that will be of inestimable value to him in his engineering practice. This should be the prelude to a close questioning on the entire ground covered by the lecture and the text. There should also be ample opportunity for explanations that may be desired by the students. In fact, a considerable portion of the lectures should be largely of the nature of conferences, so that independence of thought and a keen interest in the subject may be cultivated. A generous amount of work on problems resembling as closely as possible those arising in practice, together with full demonstrations of principles, should be constantly interspersed with the other work of the course. In these problems and demonstrations, the student should be required to defend his demonstrations against criticism. He should be trained to state and defend his views and demonstrations in a concise, clear, and forcible manner, and the attainment of that important end will require all the exercises of this character that any curriculum can afford. To the writer's mind, this matter of the best method of instruction is beset with many various difficulties, some of which can only best be solved in view of the personal qualifications of the instructor. It is a question that has at best given him years of most serious thought, both in the lecture room and in active practice, and both these plans of experience have led him by slow stages to believe that its best solution will be found consistent with the general principles above stated; but he also believes that in the application of those principles the largest liberty for the competent instructor in individual choice and action is most essential.

Such the writer deems to be the requisite features and the main characteristics of the course of study which will enable those who pursue it to acquire the “Ideal Engineering Education.” There are then to be grouped along the main lines such a selection and number of subjects as each special field of engineering activity may require, so that each course of study may be completed in all its details. It may be and probably will be objected that this “ideal” is too ideal. But such an objection is groundless; the ideal” is already too nearly accomplished. Again, aside from such considerations, it is well to remember that that which is not the object of effort is never attained.

REQUIREMENTS IN MATHEMATICS FOR ENGINEER

ING EDUCATION.

BY ARTHUR N. TALBOT, Professor of Municipal Engineering and Mechanics, University of

Illinois.

Sixteen years ago the author of an engineers’ pocket-book stated in its preface that, having long since forgotten the little higher mathematics he once knew, he could not read the profound language of Rankine, Mosely and Weisbach, and that their books are beyond the reach of all but a few engineers, adding that they are but little more than striking instances of how completely the most simple facts may be buried out of sight under heaps of mathematical rubbish. The late editions of this valuable pocket-book have been so thoroughly remodeled that its present arrangement is sufficient refutation of the implication that mathematics is useless for an engineer. Recent engineering textbooks show the transformation from the two extremes of Rankine and Trautwine to the present ideas of useful engineering mathematics. It is to this present condition of the place and amount of mathematics in the education of the engineer that your attention is invited.

In order better to consider this relation, it may be well to state the commonly recognized aims and objects of the education of the engineer. These perhaps may

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