result of this, when contracts are made according to which an engine must deliver a brake horse power for the expenditure of a definite number of heat units, it makes considerable difference, both commercially and practically, as to whether the pressure of the gas-fuel used is given in absolute or gravitational units when compared with results obtained elsewhere.

PROFESSOR A. N. TALBOT: The last remark of Professor Woodward is worthy of another word. He stated that the boy who studies physics says "g" is the force of gravity. When that is traced back it is found that the text-book ordinarily says 66 g is the attraction of gravity on the unit mass." But the boy forgets all about unit mass, and as most of the equations he happens to use in physics deal with unit mass, he gets the idea that g is a force. This is the source of much confusion.

PROFESSOR MAURER: As one device for explaining acceleration I have the student consider the velocity changes in the case of a freely falling body, using that case because he is more or less familiar with it and probably knows the equations s=16t2 and v=32t. I ask him to make a schedule of the motion, writing in the first column the numbers 0, 1, 2, 3, etc., to denote values of t, and in the next two the corresponding values of s and v. He knows that these columns should be labelled respectively as follows: "t in seconds," "s in feet," and "v in feet per second." Then he is directed to set down in the fourth and fifth columns the velocity increments for successive seconds and minutes respectively. He readily understands that these two columns should be labelled "velocity increments

in feet per second." Observing now that these increments, or velocity changes (32 and 1,920), occur per unit time, he sees that they are values of the acceleration of the falling body and also that the full description of the acceleration requires the naming of the velocity increment and the unit time per which it occurs, thus 32 feet per second per second, and 1,920 feet per second per minute.

I do not wish to be understood as antagonizing the views of anybody on these matters for I know that there are other positions than those taken by myself which are sound. But believing that more uniformity and system in our systems of units would be of advantage to students, I have ventured to present a set of views which unify the systems, and are sound and in line with the trend of good usage.

ON TEACHING CALCULUS TO ENGINEERING

STUDENTS.

BY ALFRED M. KENYON,

Professor of Mathematics, Purdue University.

It is proposed in this paper to set forth some of the methods of presentation and the principal topics presented to the engineering students of the sophomore class in Purdue University. This first course in calculus is given in the second semester of the sophomore year, having been preceded in the first semester by analytic geometry, four hours a week being assigned to each subject in the program. At the opening of the junior year the engineering students take up theoretical and applied mechanics, which continues throughout the year and is accompanied by the second course in calculus two hours a week. The mechanics is not administered by the department of mathematics.

The primary object of this course is to ground the student thoroughly in a few elementary and fundamental principles of calculus; the secondary object is to prepare him to apply these principles and to understand the application of similar ones in his work in mechanics.

In order to grasp the fundamental principles of calculus it is necessary first of all to have a clear understanding of the system of real numbers and of the operations which may, and which may not, be performed with these numbers, and of the uniqueness and definiteness of the results of these operations. With such an

( 221 )

understanding there is no necessity for any mystery or even indefiniteness attaching to such terms as limit, infinity, etc.

A new principle is presented to the student through the medium of a well-chosen example from geometry or physics which is worked out always to a definite numerical result; the exposition, proof, and formulation, in one or another order, then follow. The student then works out numerous problems chosen from physics, laboratory or engineering practice, or geometry, to fix and make definite the principle in his own mind and to get a general idea of the kind of problem in which it is likely to have an application. Great stress is laid on geometrical or graphical representation, by the use of rectangular and polar coördinate paper, and other devices, of the numbers occurring in these problems, and upon the interpretation of the successive processes and results from the graph.

A somewhat limited but consistent and convincing experience has led the instructors of the department to regard this method of presentation as very efficient, to attach especial importance to the right choice of problems, and to appreciate the definiteness that results in the mind of the student who works a considerable number of such problems to a numerical result. Such problems always precede those which (on account of the presence of arbitrary constants) lead to formulas; the latter being introduced very sparingly and gradually as the course proceeds.

As to the matter of rigor, it is not regarded a sin (either moral or logical) for a student to use knowledge which he can not prove; it is all wrong to put him under

the impression that he has proved it when he has not; it is indefensible for either the instructor or the textbook to give him "knowledge" which is not true, or to set before him for proofs, plausible statements which do not prove. It is not feasible, and it is not pedagogically wise to give rigorous proofs of all the theorems that are presented in this course; the student is not ready for it, and even if he were, the time and attention required would detract from the main issue and defeat the very objects of the course; but what is insisted on is that he shall have definite ideas of what is presented to him and shall know exactly what has been proved and what he has accepted on authority and what at some time in an advanced course in the theory of functions of a real variable he must prove in order to put his knowledge of calculus on a strictly logical basis. This is important in the first place because it is nothing more than common honesty on the part of the instructor and the text-book and in the second place because it puts the student in position to do further work in the subject without discovering that his foundations can not be depended up. The discovery that a general theorem, which he has learned and proved in calculus, is not always true, and hence under the hypotheses stated is not generally true, has often wrought confusion in the mind of a student by raising the insidious but pertinent question whether any of the proofs he has been given hold good, has thrown the calculus under suspicion and has led him to abandon it as an instrument to be applied in engineering work. Upon this point I beg to quote from Professor Maximè Bôcher in the Annals of Mathematics, vol. 2, page 81: " One of the