A College Course for Alumni

The Sophomore Course in the Calculus

THE calculus was invented by Isaac Newton, and independently by Gottfried Leibnitz about the year 1680. During the remainder of the seventeenth century, four men—Newton, Leibnitz, and the two Bernoulli brothers—were the only real masters of the subject. It is admitted that twelve men understand the Einstein theory. It would thus appear that the calculus was at least three times as difficult for its day and generation as the Einstein theory is for ours. We now teach the rudiments of the calculus to about threequarters of the freshmen, and nearly one-quarter of the sophomores elect a full year course in the calculus. What the future may have in store for Dartmouth undergraduates is a solemn thought indeed.

The calculus has always seemed to me to be professional mathematics. I wish to qualify this statement by adding that I am not sure that I know exactly what it means. It is increasingly evident that in sport the terms amateur and professional mean many things to many men. Yet these are useful terms, even though we do not always define them with precision. When we do so define them, arguments arise. But the worker in any one of the exact sciences finds that the subject matter of the calculus is a necessary part of his professional equipment; there is no argument about that. A striking editorial in The Saturday Evening Post for June 1,1929, closes with the words: "Every circumstance points to the belief that if we are to keep up with the procession of human progress our schools and colleges will have to devote more time to this subject (mathematics), offer more advanced courses and stress their importance to every student who hopes to make any real progress in the physical sciences." I do not think there is any contradiction between all this and the fact that in a college of liberal arts many men pursue a professional course in the calculus, and I believe with profit, and yet never make any use of it professionally or otherwise. Of course the calculus can be presented in an amateurish way. A number of recent books try in a light, amusing manner to show as they say "what it's all about." But inherently the calculus, though often amusing, requires a certain amount of effort, and the injection of morphine usually brings on general paralysis.

The calculus is divided into two principal parts, the differential calculus, and the integral calculus. Each has its own technique, and each has very important app lications. Historically these were developed simultaneously. The text book authors of the nineteenth century separated the subject into what modern disciples of pedagogy term water-tight compartments. Recent attempts at fusion have not been particularly successful.

RELATIVE VALUES

In scientific work, and in everyday life, we often observe that two things are related, so that a change in one produces a change in the other. Thus the distance a car travels depends on the time. A change in the distance is produced by a change in the time, and the quotient of these changes is the average speed during the interval of time. The speed at an instant is the limiting value of this average speed as the interval of time decreases indefinitely. The finding of this instantaneous speed is a typical problem of the differential calculus. The speed is called the derivative of the distance with respect to the time. More generally we define, relative to any given function, another function, called the derivative. The differential calculus concerns itself with two things, the process of finding the derivative, and the use of it after we have found it. When the students can find the derivative of any function which their instructors can invent, and these instructors have some slight ability in invention, we proceed to the applications of the derivative: rates, curve tracing, Newton's method of approximation, and maxima and minima.

Integration is the inverse of differentiation. We are given the derivative, and try to recover the function. Here the technique is vastly different. In differentiation, one straight-forward process leads to a set of fundamental formulas by means of which the derivat ive of any respectable function can be found at once. In integration there is no direct method, but instead there are substitutions, transformations, and not a few methods which can only be described as tricks of the trade. For a time students exhibit a somewhat surprising degree of humility. Parenthetically, integration is a cold weather subject; on no account should it be continued into the month of March, at least not a Hanover March. An amazing theorem now allows us to apply integration to find the length of any curve, the area of any plane or curved region, the volume of any solid, the center of gravity of any mass, the gravitat ional attraction of any mass, and many other important measures of geometry and physics. Suddenly our sophomore finds himself repeating some of the famous discoveries of Newton and Leibnitz. He gets a real thrill out of finding the area under one arch of the cycloid, a problem which neither Galileo nor any of his contemporaries could solve. The results which he now obtains make the effort and inevitable drudgery of the last five years worth while.

Algebra and geometry, honorable and respected though they may be, sometimes remind us of the sermon which Samuel Pepys records in his diary: "A good honest and painful sermon." It is true that these subjects are sometimes taught with rare appreciation and artistry. It is true that they can be taught with enormous gusto. It is still true that for many boys the work is sometimes dull and sometimes tedious. Coming to college, the boy studies analytic geometry. In this course, algebra and geometry, which up to this time had been separate and distinct disciplines, are put together, and while the primary aim of the course is the study of geometry by means of algebra, there is considerable advance in algebra through following up geometric hunches. By the end of his freshman year our young gentleman has to some extent co-ordinated these two subjects and, mathematically speaking, has learned to walk. He may even in this year surmount a few foothills, and obtain a few glimpses of the domain of scientific thought, but he is still in the woods. If he wishes a comprehensive view of this domain, he will find that the sophomore course in the calculus is a well organized party which will lead him above timber-line.

AN OLD FASHIONED WINTER (l 898)