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the way the subject is approached. I can never forget a remark once made by Professor Charles R. Cross to the effect that mathematics was nothing but common sense in shorthand. So it is. History will teach us that mathematics has been developed, not as an end in itself, but as a means to an end, namely to compress long exact arguments within such a scope that they can be readily grasped all at once within the time-span of the reader. The same end is attained in good writing where the author clearly states in one short sentence what the ordinary man could hardly express in half a page. The meaning of the sentence or of the formula is thus brought within the span of an instant of time and both its first and last word or sign are grasped at once.

We find mathematics primarily used to express briefly those factors in a process that are observed to remain constant throughout that process. A formula thus expresses the form of the invariants of a series of related facts. Hence, logically, the observation of the facts should come first and the formula follows. Two oppositely charged spheres do not follow the law of inverse squares. The law of inverse squares describes for the purpose at hand the behavior of the two oppositely charged spheres. Physics is to be taught not as a series of cases of the validity of certain formulas, but the formulas are to be found as adequately describing the conduct of nature.

PROFESSOR A. N. TALBOT: I have always been in favor of making a generous use of applications to engineering in teaching mathematics, not so much for the skill and facility in making applications which the student may acquire in this way, but because such applications for most students form a good means of giving a clear conception of mathematical ideas and processes which might otherwise be very hazy. The applications give life and meaning to seemingly abstruse mathematical statements. And yet in late years it has seemed to me that there is a tendency in this country to carry this to an extreme and for the student merely to learn how to apply rules or formulas which he assumes have already been established. Formal demonstrations and the philosophy of the work are likely to be disregarded. As the pendulum swings toward the practical side, it must not be forgotten that sound logic, formal demonstration, and mathematical form of statement are essential features in the solution of engineering problems and that the development which goes with a training along these lines should form an important element in the education of the engineering student.

PROFESSOR C. M. WOODWARD: I have always rather regretted the attempt to limit the range of mathematical study even in an engineering school. One never knows when mathematics will be of use. I believe that a comprehension of the meaning of mathematics is one of the fruits of good mathematical training. I think I stated that a certain college president regretted that mathematics was found so useful. He said that mathematics has been found so useful that its educational value has been greatly hampered. Now I thought that superb nonsense. I believe that the mastery of a subject is what makes it valuable and it cannot be thoroughly learned until you have seen that you are the master of it. And so I like to see the best kind of mathematics of service to the world at large.

PROFESSOR WALDO: Nothing can prevent the deduction of new terms. I should like to have in this paper a statement of how the term engineering mathematics comes. I do not know what engineering mathematics is and I do not see any difference between one kind of mathematics and other kind of mathematics.

PROFESSOR HENRY T. EDDY: I think that some of the great scope of mathematics has been directly suggested and stimulated by its applications. It is not unfair in our educational system that mathematics be connected with the reality. As all knowledge is essentially experience, and as everything new is that suggested to us by our own experience, or that of the race, I think that there is no real difference between pure mathematics and applied mathematics, although this distinction is frequently drawn. I love pure mathematics, but I think I love applied mathematics better.

PROFESSOR TYLER: I agree with Professor Eddy. While the term engineering is based on a good underlying idea, namely, mathematics, the adaptation of mathematical instruction to engineering uses, there is no more real reason for "engineering mathematics” as a distinct subject or title than for "engineering physics" or "engineering chemistry." The university president who Professor Woodward tells us deplored the increasing applications of mathematics was, I think, far from representing the attitude of our present day mathematicians. The use of such a title as "engineering mathematics" seems to me calculated to promote an unfortunate tendency which would leave the pure mathematics to the mathematicians and leave the applied mathematics to the engi

The increasing utilization of mathematics is in this country the great hope of the science for the



future, and should be welcomed by the mathematicians most cordially.

PROFESSOR WALDO: I think that possibly the idea which the author had in mind when he made the distinction between pure mathematics and engineering mathematics was that the latter is more thoroughly understood. If we investigate mathematical conditions prevailing in our institutions at about the seventies, we will discover that mathematics was the bugbear of the collegian to be memorized for the examination and forgotten as quickly as possible. Engineering schools have greatly changed these conditions. The person who does not wish to pursue a mathematical course takes what the academy requires of him and stops, but the person who is going to apply his mathematics seeks more in an intelligent way and in the end has something to show for his work. Of course in all of our institutions there will be a saving remnant who love mathematics and pursue it for its own sake, but their number is small; but all our great body of trained engineers must know and use a moderate amount of higher mathematics as a tool with which to serve mankind. It is their useful work that has rescued the pure mathematician from the derision of his fellows.

PROFESSOR SWAIN: I wish to emphasize one point which seems to me open to criticism, and that is the idea that there should be a different kind of teaching of mathematics in an engineering school from that which is suitable in colleges. It seems to me that there is only one way to teach mathematics. The main object of all teaching is to give the student some power of thought, some power to do something which he could not do before. I, therefore, think that there is just one way to teach mathematics, and that is the way that we try to teach it to engineering students so that they will be able to use it when they have learned it. Unfortunately, much teaching is not of that kind and gives no power. I do not agree with the idea that mathematics taught for general culture, so-called, should be taught any differently from teaching it with the use of it in view, and I hope that the idea of teaching mathematics, or anything else, for that matter, with its applications in view will come into use more and more. I think that the teacher of mathematics who has been trained as an engineer, or in some branch of applied science, so that he knows the uses of the tools he is teaching, will have a great advantage as a teacher, and achieve much better results than a man who, though he may be a fine mathematician, is an abstract one.

MR. B. JONES, JR.: During the last four or five years a course of lectures to journeyman electrical workers and operating steam engineers has been given at the New York Trade School. These lectures were primarily for men engaged in handling and erecting steam and electrical apparatus and who were anxious to better themselves. The fundamental idea at the basis of the course was to teach these men to think.

Generally the men who attended were bright and able. They only required to be showed “how," but like most men of their class they had an inordinate dread of theory and of mathematics. This was overcome very simply by showing by means of experiments that theory and formulas are no more than descriptions of fundamental physical facts. Thus the law of the magnetic circuit was taught by means of

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