I am not prepared to say that we have arrived at such a critical point in the development of educational systems, but I certainly think that the relatively sudden and deep interest that has been recently shown in the subject points to the near approach of such a point.

THE JUSTIFICATION OF THE USE OF THE

66

EXPRESSION ENGINEERING

MATHEMATICS."

BY ARTHUR E. HAYNES,

Professor of Engineering Mathematics, University of Minnesota.

I am constrained to write briefly on the above topic because, so far as I am aware, the expression "Engineering Mathematics" originated in the University of Minnesota in January, 1901, when the writer, with the approval of the president of the university, changed the name of the professorship he occupied from the professorship of mathematics in the college of engineering to the professorship of engineering mathematics.

The reasons for the use of this name are at least three.

First: Because of the main object of the study of mathematics in an engineering course. In a purely disciplinary or academic course, the objects of the study are, first, mental discipline and, second, its practical use. In a professional course, such as engineering, these two objects are reversed in relative importance, and the primary object in such a course is to so master the subject as to be able to make a correct and intelligent use of its principles and formulas in practice. In other words, these principles and formulas are to be used as "tools"* by which, like as the student uses a plane or any other tool of the shop, he may secure desired results.

*I think that this expression was first used in this university. See Proceedings of the Society for the Promotion of Engineering Education, Vol. VIII., pp. 308-310. [This is questioned.-W. T. M.]

Incidentally, in mastering the logic involved, the student gets excellent mental discipline, but there is added to this a still greater discipline and a commendable inspiration in applying his knowledge so as to get direct practical results.

There is, to be sure, some valuable discipline in so studying a jack plane as to understand its construction completely, but one can do all this without being able to properly plane a board.

Just so one may, for instance, be able to develop a mathematical principle or formula without being able to use it intelligently in securing results.

Another example is ventured for illustration; in physics, where the members of the class are simply studying for the knowledge alone, and not for the object of applying such knowledge, they may completely understand the philosophy, for instance, of the sextant, and yet be perfectly helpless when required to find the angular distance between two objects by means of its use.

In fact, in the study of mathematics of an engineering course, the student must constantly challenge principles and formulas with the question what use has this in practical application to my chosen profession. In truth, three questions are constantly pressing upon both the student of engineering and the professional engineer for positive and definite answers; they are: What do I want? Do I know how to secure it? Do I know how to use it intelligently and correctly?

Suppose, for example, he finds it necessary to solve a problem in trigonometry, he may have to answer these questions repeatedly, as in determining the proper formulas, and in correctly selecting and apply

ing them; again, in the same problem, the determining whether to use natural or logarithmic functions and the proper selection and correct use of them. And thus it comes about, in the study of mathematics for engineering purposes, that the main object sought fixes or shapes the method of study. But this leads naturally to another reason for the justification of the expression and that is:

Second: Because such an object requires a different kind of teaching. Some one has recently said: “In almost every branch of instruction there is a gap between theory and practice, between principle and application. In nothing is this more manifest than in the teaching of mathematics."

As has been already indicated, one may, by mastering a mathematical principle, secure valuable mental discipline; but if the teacher makes this the primary or chief object, he will not teach the subject with the success that he would have, by having constantly before him the thought that skill in the intelligent use of mathematics in practice is the main end to be attained.

And so the question may here be legitimately asked, will the teacher untrained in engineering practice or not educated as an engineer, be as likely to succeed in such teaching as one who is so trained or educated? I think this question must be, in general, answered in the negative.

Such teaching requires, on the part of the teacher, that he be able to intelligently direct the student in his study, that he have the ability and good judgment to select problems that shall wisely lead the student to a correct and practical application of his knowledge, and above all, that he possess the power of con

stantly inspiring the pupil by high ideals of work and of life.

But if the object sought shapes both the method of study and of the teaching of the subject, it leads directly and logically to still another reason for the justification of the expression, and so we have:

Third: Because the content or extent of the field covered is different. For example, long and abstruse discussions of principles and formulas are omitted, the object of which is mostly mental discipline and which have little or no direct practical value, while subjects not found in former treatises are added. A good illustration of this is seen in introducing the philosophy and use of the slide rule after the theory and use of ordinary logarithms have been properly considered. The student is thus led to the intelligent comprehension and use of one of the most valuable aids in modern engineering practice.

Again, after a study of the theory of ordinary integration, the study and use of the polar planimeter, or mechanical integrator, adds great interest and value to the student's work.

In conclusion, it may be said that, in consequence of the three reasons given in this paper for the justification of the term "engineering mathematics," great and valuable changes have been made during the last decade in mathematical text-books for engineering students, and we may ultimately expect nothing further to be desired in such works.*

DISCUSSION.

MR. B. JONES, JR.: A great part of the trouble with our present methods of teaching mathematics lies in *I think the use of the terms Engineering Drawing, Engineering Physics, etc., is equally justifiable.—A. E. H.