Disciplinary value of the study

If culture means the subjective transformation of information into a philosophy of life, can culture be complete unless it has included in its reflections the marvelously simple yet intricate interrelations of natural phenomena? The value of this intricate simplicity as a mental discipline is equaled perhaps only in the finely drawn distinctions of philosophy and in the painstaking statements of limitations and the rapid generalizations of pure mathematics; and let us not forget the value of discipline, outgrown and unheeded though it be in the acquisitive life of the present age.

Relation of physics to philosophy and the exact sciences

The professional student, continually increasing in numbers in our colleges, either of science or in certain branches of law, finds a broad familiarity with the latest points of view of the physicist not only helpful but often indispensable. Chemistry can find with difficulty any artificial basis for a boundary of its domain from that of physics. Certainly no real one exists. The biologist is heard asking about the latest idea in atomic evolution and the electrical theories of matter, hoping to find in these illuminating points of view, he tells us, some analogy to his almost hopelessly complex problems of life and heredity. Even those medical men whose interest is entirely commercial appreciate the convenience of the X-ray and the importance of correctly interpreting the pathological effects of the rays of radio-activity and ultra-violet light. One finds a great geologist in collaboration with his distinguished colleague in physics, and from the latter comes a contribution on the rigidity of the earth. Astronomy answers nowadays to the name of astrophysics, and progressive observatories recognize in the laboratory a tool as essential as the telescope. In a word, the professional student of science not only finds that the subject matter of physics has many fundamental points of contact with his own chosen field, but also recognizes that the less complex nature of its material allows the method of study to stand out in bolder relief. Training in the method and a passion for the method are vital to a successful and an ardent career.

Should the teaching of college physics change its aim for different classes of students?

In the teaching of physics, then, the aim might at first sight appear to be quite varied, differing with different classes of students. A careful analysis of the situation, however, will show, we think, that this conclusion can with difficulty be justified: that it is necessary to conduct college instruction in a fashion dictated almost not at all by the subsequent aims of the students concerned. In the more elementary work, certainly, adherence to this idea is of great importance. The character, design, and purpose of an edifice do not appear in the foundations except that they are massive if the structure is to be great.

Not infrequently this seems an unnecessary hardship to a professional student anxious to get into the work of his chosen field. If such is the case, let him question perhaps whether any study of physics should be attempted, as this query may have different answers for different individuals. But if he is to study it at all, there is but one place where the analysis of physical phenomena can begin, and that is with fundamentals—space, time, motion, and inertia. How can one who is ignorant of the existence and characteristics of rotational inertia understand a galvanometer? How can waves be discussed unless in terms of period, amplitude, frequency, and the like, that find definition in simple harmonic motion? How does one visualize the mechanism of a gas, unless by means of such ideas as momentum interchange, energy conservation, and forces of attraction?

Let us emphasize here, lest we be misunderstood, that we are considering collegiate courses. We do not doubt that descriptive physics may be given after one fashion to farmers, quite differently to engineers, and from still a third point of view to medical students. Unfortunately some collegiate courses never get beyond the high school method. Our aim is not to discuss descriptive courses, but those that approach the subject with the spirit of critical analysis, for these alone do we deem worthy of a place in the college curriculum.

The course in college physics differentiated from the high school course

The problem of the descriptive course is the problem of the high school. Because of failure there, too often we see at many a university courses in subfreshman physics. These are made necessary where entrance requirements do not demand this subject and where subsequent interest along related lines develops among the students a tardy necessity of getting it. From the point of view of the collegiate course it often appears as if the subfreshman course could be raised to academic rank. This is because familiarity with the material must precede an analysis of it. Credit for high school physics on the records of the entrance examiner, unless this credit is based on entrance examination, is often found to stand for very little. Consequently the almost continual demand for the high school work under the direct supervision of a collegiate faculty. The number of students who should go into this course instead of the college course is increasing at the present time in the immediate locality of the writer.

As contributory testimony here, witness the number of colleges that do not take cognizance at all of high school preparation and admit to the same college classes those who have never had preparatory physics with those who have had it. We are told the difference between the two groups is insignificant. Perhaps it is. If so, this fact reflects as much on the college as on the high school. If we are looking for a solution of our problem in this direction, let us be undeceived; we are looking backwards, not forward.

Need of adequate high school preparation in physics

No one will affirm that to a class of whose numbers some have never had high school physics a course that is really analytical can be given. Wherever a rigorous analytic course is given those who have been well trained in descriptive physics do well in it in general. Let us not beg the question by giving such physics in a college that does not require high school preparation. The college curriculum is full enough as it is without duplication of high school work, and any college physics course that is a first course is essentially a high school course.

Let us rather put the responsibility squarely where it lies. The high school will respond if the urgency is made clear. Witness some of them in our cities already attempting the junior college idea, an idea that has not been unsuccessful in some of our private schools. If it is made clear that a thoroughgoing course in descriptive physics is a paramount necessity in college work and that no effort will be spared on the part of the university to insure this quality, the men will be found and the proper courses given.

Preparatory work in mathematics essential for success in college physics

We favor a comprehensive examination plan in all cases where the quality of the high school work is either unknown or open to question.

Familiarity, likewise, with the most elementary uses of mathematics should be insured. It would be highly desirable that a course of collegiate grade in trigonometry should immediately precede the physics. This is not because the details of trigonometry are all needed in physics. In fact, a few who have never had trigonometry make a conspicuous success in physics. These, however, are ones who have a natural facility in analysis. To keep them out because of failure to have had a prerequisite course in trigonometry often works an unnecessary hardship. We would argue, therefore, for a formal prerequisite on this subject, reserving for certain students exemption, which should be determined in all cases, if not by the instructor himself, at least by his cooperation with some advisory administrative officer.

Need of testing each student's preparation

Nor is it sufficient with regard to the mathematical preparation or the knowledge of high school physics in either case to go exclusively by the official credit record of the student. It is our firm conviction from several years' experience where widely different aims in the student body are represented that above and beyond all formal records attention to the individual case is of prime importance. The opening week of the course should be so conducted that those who are obviously unequipped can be located and directed elsewhere into the proper work. How this may best be accomplished can be determined only by the circumstances in the individual school, we imagine. Daily tests covering the simplest descriptive information that should be retained from high school physics and requiring the intelligent use of arithmetic, elementary algebra, and geometry will reveal amazing incapacity in these things. Tuttle, in his little book entitled An Introduction to Laboratory Physics (Jefferson Laboratory of Physics, Philadelphia, 1915), gives on pages 15-16 an excellent list of questions of this sort. Any one with teaching experience in the subject whatever can make up an equally good one suited for his special needs and temperament. It should not be assumed that all who fail in such tests should be dropped. Some undoubtedly should be sent back to high school work or its equivalent; others may need double the required work in mathematics to overcome their unreadiness in its use. Personal contacts will show that some are drifting into a scientific course who have no aptitude for it and who will be doomed to disappointment should they continue. In a word, then, we are convinced that the more carefully one plans the work of the first week or so the more smoothly does the work of the rest of the year follow. The number of failures may be reduced to a few per cent without in any way relaxing the standard of the course.

Methods of teaching college physics

With regard to the organization of the college courses in physics there seems to us to be at least one method that leads to a considerable degree of success. This is not the lecture method of instruction; neither is it a wholly unmitigated laboratory method.

Lecture method vs. laboratory method

To kindle inspiration and enthusiasm nothing can equal the contact in lectures with others, preferably leaders in their profession, but at least men who possess one of these qualities. Such contacts need not be frequent; indeed, they should not be. The speaker is apt to make more effort, the student to be more responsive, if such occasions are relatively rare. Even thus, although real information is imparted at such a time, it is seldom acquired. However, perspective is furnished, interest stimulated, and the occasion enjoyed.

Limitations of exclusive use of each method

For the real acquisition of scientific information, the great method is the working out of a laboratory exercise and pertinent problems, with informal guidance in the atmosphere of active study and discussion engendered among a small group,—the laboratory method. Taken alone, it is apt to become mechanical and uninteresting and the outlook to be obscured by details. Lectures, especially demonstration lectures, are needed to vitalize and inspire. Moreover, many of the most vivid illustrations of physical principles that occur on every hand to focus the popular attention are never met with in the college course because they are unsuited for inexperienced hands or not readily amenable to quantitative experimentation. The more informally such demonstrations can be conducted, the more enthusiastically they are received.

Aims of the laboratory method

With regard to laboratory work, accuracy in moderate degree is important, but too great insistence upon it is apt to overshadow the higher aim; namely, that of the analysis of the phenomena themselves. A determination of the pressure coefficient of a gas to half a per cent, accompanied by a clear visualization of the mechanism by which a gas exerts a pressure and a usable identification of temperature with kinetic agitation, would seem preferable to an experimental error of a tenth per cent which may be exacted which is unaccompanied by these inspiring and rather modern points of view. Especially in electricity is a familiarity with the essentials of the modern theories important. Here supplementary lectures are of great necessity, for no textbook keeps pace with progress in this tremendously important field. Problem solving with class discussion is absolutely essential, and should occupy at least one third of the entire time. In no other way can one be convinced that the student is doing anything more than committing to memory, or blindly following directions with no reaction of his own.

Value of the supplementary lecture

The incorporation recently of this idea into the courses at the University of Chicago has been very successful. Five sections which are under different instructors are combined one day a week at an hour when there are no other university engagements, for a lecture demonstration. This is given by a senior member of the staff whenever possible. The other meetings during the week are conducted by the individual instructors and consist of two two-hour laboratory periods and two class periods that usually run into somewhat over one hour each. These sections are limited to twenty-five, and a smaller number than this would be desirable. The responsibility for the course rests naturally upon the individual instructors of these small sections. These men also share in the demonstration work, since each is usually an enthusiast in some particular field and will make a great effort in his own specialty to give a successful popular presentation of the important ideas involved. The enthusiasm which this plan has engendered is very great. Attendance is crowded and there is always a row of visitors, teachers of the vicinity, advanced students in other fields of work, or undergraduates brought in by members of the class. These latter especially are encouraged, as this does much to offset current ideas that physics is a subject of unmitigated severity. The particular topics put into these demonstrations will be discussed in paragraphs below, which take up in more detail the organization of the special subdivisions of the material in a general physics course.

Mechanics a stumbling block—How to meet the difficulty

Mechanics is a stumbling block at the outset. As we have indicated above, it must form the beginning of any course that is analytic in aim. There is no question of sidestepping the difficulty: it must be surmounted. A judicious weeding during the first week is the initial part of the plan. Interest may be aroused at once in the demonstration lectures by mechanical tricks that show apparent violations of Newton's Laws. These group around the type of experiment which shows a modification of the natural uniform rectilinear motion of any object by some hidden force, most often a concealed magnetic field. The instinctive adherence of every one to Newton's dynamic definition, that acceleration defies the ratio of force to inertia, is made obvious by the amusement with which a trick in apparent defiance of this principle is greeted. Informality of discussion in such experiments, questions on the part of the instructor that are more than rhetorical, and volunteer answers and comment from the class increase the vividness of the impressions. A mechanical adaptation of the "monkey on the string" problem, using little electric hoists or clockworks, introduces interesting discussion of the third law in conjunction with the second. A toy cannon and target mounted on easily rolling carriages bring in the similar ideas where impulses rather than forces alone can be measured.

There follow, then, the laboratory experiments of the Atwood machine and the force table, where quantitative results are demanded. It is desirable to have these experiments at least worked by the class in unison. Whatever may be the exigencies of numbers and apparatus equipment that prevent it later, these introductions should be given to and discussed by all together. In the nature of things, fortunately, this is possible. A single Atwood machine will give traces for all in a short time under the guidance of the instructor. The force table experiment is nine-tenths calculation, and verifications may be made for a large number in a short time. Searching problems and discussion are instigated at once, and the notion of rotational equilibrium and force moments brought in. Because of the very great difficulty seeming to attach to force resolutions, demonstration experiments and problems using a bridge structure, such as the Harvard experimental truss, will amply repay the time invested. Another experiment here, which makes analysis of the practice of weighing, is possible, although there will be divergence almost at once due to the personality of the instructor and the equipment by which he finds himself limited. The early introduction of moments is important, however, because it seems as if a great amount of unnecessary confusion on this topic is continually cropping out later. At this point, if limitations of apparatus present a difficulty, a group of more or less independent experiments may be started. Ideas of energy may be illustrated in the determination of the efficiency and the horse power of simple machines, such as water motors, pulleys, and even small gas or steam engines.

In discussion of power one should not forget that in practical problems one meets power as force times velocity rather more frequently than as rate of doing work, and this aspect should be emphasized in the experiments. Conservation of energy is brought out in these same experiments with reference to the efficiencies involved. In sharp contrast here the principle of conservation of momentum may be brought in by ballistic pendulum experiments involving elastic and inelastic impacts. Most students are unfamiliar with the application of these ideas to the determination of projectile velocities, and this forms an interesting lecture demonstration. Elasticity likewise is a topic that may be introduced with more or less emphasis according to the predilection of the instructor. The moduli of Young and of simple rigidity lend themselves readily to quantitative laboratory experiments. Any amount of interesting material may be culled here from recent investigations of Michelson, Bridgman, and others with regard to elastic limits, departures from the simple relations, variations with pressure, etc., for a lantern or demonstration talk in these connections.

By this time the student should have found himself sufficiently prepared to take up problems of rotational motion. The application of Newton's Laws to pure rotations and combinations of rotation and translation, such as rolling motions, are very many. We would emphasize here the dynamic definition of moment of inertia, I = Fh/a rather than the one so frequently given importance for computational purposes, Smr^{2}. Quantitative experiments are furnished by the rotational counterpart of the Atwood machine. Lecture demonstrations for several talks abound: stability of spin about the axis of greatest inertia, Kelvin's famous experiments with eggs and tops containing liquids, which suggest the gyroscopic ideas, and finally a discussion of gyroscopes and their multitudinous applications. The book of Crabtree, Spinning Tops and the Gyroscope, and the several papers by Gray in the Proceedings of the Physical Society of London, summarize a wealth of material. If one wishes to interject a parenthetical discussion of the Bernouilli principle, and the simplest laws of pressure distributions on plane surfaces moving through a resisting medium, a group of striking demonstrations is possible involving this notion, and by simple combination of it with the precession of a rotating body the boomerang may be brought in and its action for the major part given explanation.

Rotational motion leads naturally to a discussion of centripetal force, and this in turn is simple harmonic motion. This latter finds most important applications in the pendulum experiments, and no end of material is here to be found in any of the textbooks. The greatest refinement of experimentation for elementary purposes will be the determination of "g" by the method of coincidences between a simple pendulum and the standard clock. Elementary analysis without use of calculus reaches its culmination in a discussion of forced vibrations similar to that used by Magie in his general text. Many will not care to go as far as this. Others will go farther and discuss Kater's pendulum and the small corrections needed for precision, for here does precision find bold expression.

It is not our purpose to give a synopsis of the entire general physics course. We have made an especially detailed study of mechanics, because this topic is the one of greatest difficulty by far in the pedagogy. It is too formally given in the average text, and seems to have suffered most of all from lack of imagination on the part of instructors.

Suggested content for the study of phenomena of heat and molecular physics

In the field of heat and molecular physics in general there is much better textbook material. Experiments here may legitimately be called precise, for the gas laws, temperature coefficients, and densities of gases and saturated vapor pressures will readily yield in comparatively inexperienced hands an accuracy of about one in a thousand. In the demonstrations emphasis should be given to the visualization of the kinetic theory points of view. Such models as the Northrup visible molecule apparatus are very helpful. However, in absence of funds for such elaboration, slides from imaginative drawings showing to scale conditions in solids, liquids, and vapors with average free paths indicated and the history of single molecules depicted will be found ideal in getting the visualization home to the student. Where we have a theory so completely established as the mechanical theory of heat it seems quite fair to have recourse to the eye of the senses to aid the eye of the mind. Brownian movements have already yielded up their dances to the motion picture camera. Need the "movies" be the only ones to profit by the animated cartoon?

Nor should the classical material be forgotten. Boys' experiments in soap bubbles have been the inspiration of generations of students of capillarity. And if the physicist will consult with the physiological chemist he will find a mass of material of which he never dreamed where these phenomena of surface tension enter in a most direct fashion to leading questions in the life sciences.

The teacher of scholarship and understanding is the teacher who uses sound methods

Enough has been said to indicate what we consider the methods of successful teaching of college physics. It is quite obvious, we think, that physics constitutes no exception to the rule that the teacher must first of all know and understand his subject. Right here lies probably nine tenths of the fault with our pedagogy. No amount of study of method will yield such returns as the study of the subject itself. The honest student, and every teacher should belong to this class or he has no claim to the name, is well aware that most of his deficiency in explaining a topic is in direct ratio to his own lack of comprehension of it. In physics, as in every other walk of life, we suffer from lack of thoroughness, from a kind of superficiality that is characteristically human but especially American. We have yet to know of any one who really ranks as a scholar in his subject from whom students do not derive inspiration and enthusiasm. Such a one usually pays little attention to the methods of others, for the divine fire of knowledge itself does not need much of tinder to kindle the torches of others. Our greatest plea is for our teachers to be men of understanding, for then they will be found to be men of method.

The method of analysis dominant in physics

The sequence in which heat, electricity, sound, and light follow mechanics seems quite immaterial. Several equally logical plans may be organized. Preference is usually accorded one or the other on the basis of local conditions of equipment, and needs little reference to pedagogy. If one gives to mechanics its proper importance, the difficulty in giving instruction in the other topics seems very much less. The momentum acquired seems to serve for the balance of the year. Always must analysis be insisted upon, if our college course is going to differ from that of the high school. If we are to let students be content to read current from an ammeter with a calibrated scale and not have the interest to inquire and the ambition to insist upon the knowledge of how that calibration was originally made, we have no right to claim any collegiate rank for our courses. But if we define electrical current in terms of mechanical force which exhibits a balanced couple on a system in rotational equilibrium, there can be no dodging of the issue, for in no other way than by the study of the mechanics of the situation can the content and the limitations of our definition be understood. Any college work, so called, that does less than analyze thus is nothing more than a review and amplification of the material that should be within the range of the high school student and in that place presented to him. The first college course reveals a different method, the method of analysis. Science at the present time is so far developed that in no branch is progress made by mere description and classification. The method of analysis is dominant in the biological and the earth sciences as well as in the physics and chemistry of today.

Teaching of advanced courses in physics

On the more advanced college courses which follow the general physics course little comment is needed. Problems and questions here also exist, but they have a strongly local color and are out of place in a general discussion. The student body is no longer composed of the rank and file, half of whom are driven, by some requirement or other, into work in which they have but a passing interest at best. It is no longer a problem of seeing how much can be made to adhere in spite of indifference, of how firm a foundation can be prepared for needs as yet unrecognized in the subject of the effort. A very limited number, comparatively, enter further work of senior college courses, and these have either enthusiasm or ability and often both. Of course, a cold neglect or bored indifference in the attitude of the teacher will be resented. It will kill enthusiasm and send ability seeking inspiration elsewhere. But any one who is fond of his subject, and of moderate ability and industry, should have no difficulty in developing senior college work. If our instructor in the general course must be a scholar to be successful, the man in more advanced work must be one a fortiori. If he is not, few who come in contact with him have so little discernment as to fail to recognize the fact.

Organization of advanced courses

Organization of senior college work may be in many ways. One method where an institution follows the quarter system is the plan of having eight or ten different and rather unrelated twelve-week major courses which may be taken in almost any order. Half of these are lecture courses, the other half exclusively laboratory courses. There should be a correspondence of material to some extent between the two. Lectures on the kinetic theory of gases should have a parallel course in which the classical experiments of the senior heat laboratory are performed,—such experiments, for example, as vapor density, resistance and thermocouple pyrometry, bomb calorimetry viscosity, molecular conductivity, freezing and boiling points, recalescence, etc. A course of advanced electrical measurements should have a parallel lecture course in which the theoretical aspects of electromagnetism, the classical theories, and the equations that represent transitory and equilibrium conditions in complex circuits are discussed. In optics, likewise, there is ample material of great importance: physical, geometrical optics, spectroscopy, photography, X-ray crystallography, etc. The advanced student in these fields finds more elasticity and opportunity for cultivating a special interest in having a large number of limited interest courses from which to choose than in having such material presented in a completely organized course covering one or two years of complete work. Instructors who are specialists have opportunity of working up courses in their own fields which they do more efficiently under this plan. Research begins at innumerable places along the way, and the senior college courses so organized are the feeders of all graduate work.

Dangers of formalizing methods of instruction

In all of the above discussion it should be clearly remembered that no single plan or no one particular method has the final word or ever will have. As long as a science is growing and unfinished, points of view will continually be shifting. We are largely orthodox in our teaching. If brought up on the laboratory method of instruction it may seem the best one for us, but others may prefer another way which they have inherited. Let us appeal, then, for a constructive orthodoxy. Let us be as teachers of a subject to which we are devoted, truly and sincerely open-minded, quick to recognize and sincere in our efforts to adopt what is better wherever we meet it: waiting not to meet it, either, but going out to seek it. From the humblest college to the greatest university we shall find it here and there. Not alone in schools but in the legion of human activities about us on every hand are people who are doing things more efficiently, more thoroughly, and more skillfully than we do things. If we would be of the number that lead, we must be among the first to recognize these facts and profit by them.

First, let our work be organized with respect to that of others—the high schools; not discounting their labor but having them truly build for us.

Second, let us be open-minded enough to see that all methods of instruction have their advantages and make such combinations of the best elements in each as best suit our purpose.

Above all things, let us know our subject. Here is a task before which we quail in this generation of vast vistas. But there is no alternative for us. No amount of method will remove the curse of the superficially informed. Let us devote ourselves to smaller fields if we must, but let us not tolerate ignorance among those who bear the burden of passing on, with its flame ever more consuming, the torch of knowledge.

HARVEY B. LEMON University of Chicago



VII

THE TEACHING OF GEOLOGY

Values of the study of geology diverse

So wide is the scope of the science of the earth, so varied is its subject matter, and so diverse are the mental activities called forth in its pursuit, that its function in collegiate training cannot be summed up in an introductory phrase or two. Geology is so composite that it is better fitted to serve a related group of educational purposes than a single one alone. Besides this, these possible services have not yet become so familiar that they can be brought vividly to mind by an apt word or phrase; they need elaboration and exposition to be valued at what they are really worth. Geology is yet a young science and still growing, and as in the case of a growing boy, to know what it was a few years ago is not to know what it is today. Its disciplines take on a realistic phase in the main, but yet in some aspects appeal powerfully to the imagination. Its subject matter forms a constitutional history of our planet and its inhabitants, but yet largely wears a descriptive or a dynamic garb.

Geology a study of the process of evolution

Though basally historical, a large part of the literature of geology is concerned with the description of rocks, structural features, geologic terrains, surface configurations and their modes of formation and means of identification. A notable part of the text prepared for college students relates primarily to phenomena and processes, leaving the history of the earth to follow later in a seemingly secondary way. This has its defense in a desire first to make clear the modes of the geologic processes, to the end that the parts played by these processes in the complexities of actions that make up the historical stages may be better realized. This has the effect, however, of giving the impression that geology is primarily a study of rocks and rock-forming processes, and this impression is confirmed by the great mass of descriptive literature that has sprung almost necessarily from the task of delineating such a multitude of formations before trying to interpret their modes of origin or to assign them their places in the history of the earth. The descriptive details are the indispensable data of a sound history, and they have in addition specific values independent of their service as historical data. But into the multiplicity and complexity of the details of structure and of process, the average college student can wisely enter to a limited extent only, except as they form types, or appear in the local fields which he studies, where they serve as concrete examples of world-forming processes.

Disciplinary worth of study of geology

The study of these structures, formations, configurations, and processes yields each its own special phase of discipline and its own measure of information. The work takes on various chemical, mechanical, and biological aspects. As a means of discipline it calls for keenness and diligence in observation, circumspection in inference, a judicial balancing of factors in interpretation. An active use of the scientific imagination is called forth in following formations to inaccessible depths or beneath areas where they are concealed from view.

While thus the study of structures, formations and configurations constitutes the most obtrusive phase of geologic study and has given trend to pedagogical opinion respecting its place in a college course, such study is not, in the opinion of the writer, the foremost function of the subject in a college curriculum that is designed to be really broad, basal, and free, in contradistinction to one that is tied to a specific vocational purpose.

This study concerned primarily with the typical college course, not with vocational courses

While we recognize, with full sympathy, that the subject matter of geology enters vitally into certain vocational and prevocational courses, and, in such relations, calls for special selections of material and an appropriate handling, if it is to fulfill these purposes effectively, this seems to us aside from the purpose of this discussion, which centers on typical college training—training which is liberal in the cosmic sense, not merely from the homocentric point of view.

Knowledge of geology contributes to a truly liberal education

To subserve these broader purposes, geology is to be studied comprehensively as the evolution of the earth and its inhabitants. The earth in itself is to be regarded as an organism and as the foster-parent of a great series of organisms that sprang into being and pursued their careers in the contact zones between its rigid body and its fluidal envelopes. These contact zones are, in a special sense, the province of geography in both its physical and its biotic aspects. The evolution of the biotic and the psychic worlds in these horizons is an essential part of the history of the whole, for each factor has reacted powerfully on the others. An appreciative grasp of these great evolutions, and of their relations to one another, is essential to a really broad view of the world of which we are a part; it is scarcely less than an essential factor in a modern liberal education.

Geology embraces all the great evolutions

Let us agree, then, at the outset, that a true study of the career of the earth is not adequately compassed by a mere tracing of its inorganic history or an elucidation of its physical structure and mineral content, but that it embraces as well all the great evolutions fostered within the earth's mantles in the course of its career.

Greatest among these fostered evolutions, from the homocentric point of view, are the living, the sentient, and the thinking kingdoms that have grown up with the later phases of the physical evolution. It does not militate against this view that each of these kingdoms is, in itself, the subject of special sciences, and that these, in turn, envelop a multitude of sub-sciences, for that is true of every comprehensive unit. Nor is it inconsistent with this larger view of the scope of geology that it is, itself, often given a much narrower definition, as already implied. In its broader sense, geology is an enveloping science, surveying, in a broad historical way, many subjects that call for intensive study under more special sciences, just as human history sweeps comprehensively over a broad field cultivated more intensively by special humanistic sciences. In a comprehensive study of the earth as an organism, it is essential that there be embraced a sufficient consideration of all the vital factors that entered into its history to give these their due place and their true value among the agencies that contributed to its evolution. A true biography of the earth can no more be regarded as complete without the biotic and psychic elements that sprang forth from it, or were fostered within its mantles, than can the biography of a human being be complete with a mere sketch of his physical frame and bodily growth. The physical and biological evolutions are well recognized as essential parts of earth history. Although the mental evolutions have emerged gradually with the biological evolutions, and have run more or less nearly parallel with them—have, indeed, been a working part of them—they have been less fully and frankly recognized as elements of geological history. They have been rather scantily treated in the literature of the subject; but they are, none the less, a vital part of the great history. They have found some recognition, though much too meager, in the more comprehensive and philosophical treatises on earth-science. It may be safely prophesied that the later and higher evolutions that grace our planet will be more adequately emphasized as the science grows into its full maturity and comes into its true place among the sciences. It is important to emphasize this here, since it is preeminently the function of a liberal college course to give precedence to the comprehensive and the essential, both in its selection of its subject matter and in its treatment of what it selects. It is the function of a liberal course of study to bring that which is broad and basal and vital into relief, and to set it over against that which is limited, special, and technical, however valuable the latter may be in vocational training and in economic application.

Physical and dynamic boundaries of geology—Implications for teaching

In view of these considerations—and frankly recognizing the inadequacies of current treatment—let us note, before we go further, what are the physical and dynamic boundaries of the geologic field, that we may the better see how that field merges into the domains of other sciences. This will the better prepare us to realize the nature of the disciplines for which earth-science forms a suitable basis, as well as the types of intellectual furniture it yields to the mind. Obviously these disciplines and this substance of thought should determine the place of the science in the curriculum of any course that assumes the task of giving a broad and liberal education.

Earth-science is the domestic chapter of celestial science. Our planet is but a modest unit among the great celestial assemblage of worlds; but, modest as it is, it is that unit about which we have by far the fullest and most reliable knowledge. The earth not only furnishes the physical baseline of celestial observation, but supplies all the appliances by which inquiry penetrates the depths of the heavens. Not alone earth-science, as such, but several of the intensive sciences brought into being through the intellectual evolutions that have attended the later history of the earth, have been prerequisites to the development of the broad science of the outer heavens. The science of the lower heavens is a factor of earth-science in the definition we are just about to give. At the same time, the whole earth, including the lower heavens, is enveloped by the more comprehensive domain of celestial science.

If we seek the most logical limit that may be assigned the realm of earth-science, as distinguished from that of celestial science, of which it is the home unit, it may be found at that borderline within which any passive body obeys the call of the earth, as against the call of the outer worlds, and without which such a passive body obeys the call of the outer worlds, the call of the sun in particular. This limit is the dynamic dividing line between the kingdom of the earth and the kingdom of the outer heavens. This boundary, according to Moulton, incloses a spheroid whose minimum radius is about 620,000 miles, and whose maximum radius is about 930,000 miles. We may, then, conveniently say that the earth's sphere of control stretches out a million kilometers from its center and that this defines its true realm. At the same time, this defines the logical limit of the earth's ultra-atmosphere and appears to mark a zone of exchange between the ultra-atmosphere of the earth and the ultra-atmosphere of the sun. It thus appears to imply the place and the mode of an exchange of vital elements upon which probably hangs the wonderful maintenance of the earth's atmosphere for many millions of years and the equally wonderful regulation of the essential qualities of the atmosphere so that these have always remained within the narrow range subservient to terrestrial life. It is needless to add that this regulation also conditions the present intellectual status of the thinking factor among the inhabitants of the earth out of which—may I be pardoned for saying?—has grown the present educational discussion.

If this last shall seem to squint toward special pleading, let it be considered that, as we see things, it is precisely those views that take hold of the issues upon which our very being and all its activities depend, that serve best to train youth to broad views and penetrating thought. Such thinking seems to me to form the very essence of a really liberal education.

Not only is this definition of the sphere of geology comprehensive, but it has the special merit of being dynamic, rather than material. Such a dynamic definition comports with the view that earth-study should center on the forces and energies that actuated its evolution, since these are the most vital feature of the evolution itself. It is important to form adequate concepts of the energies that have maintained the past ongoings of the earth not only, but that still maintain its present activities and predetermine its future. It is the study of the forces and the processes of past and of present evolutions that constitute the soul of the science, rather than the apparently fixed and passive aspects of the earth's formations and configurations which are but the products of the processes that have gone before. Even the apparent passiveness of the geologic products is illusive, for they are in reality expressions of continued internal activities of an intense, though occult, order. These escape notice largely because they are balanced against one another in a system of equilibrium which pervades them and gives them the appearance of fixity. To serve their proper functions as sources of higher education, the concepts of the constitution of the earth should penetrate even to these refined aspects of physical organization and should bring the whole into harmony with the most advanced views of the real nature of physical organisms. This removes from the whole terrestrial organism every similitude of inertness and gives it a fundamental refinement, activity, and potency of the highest order. To form a true and consistent concept, the enveloping earth-science must be assumed to embrace, potentially at least, the essentials of all that was evolved within it and from it, with, of course, due recognition of what was added from without.

The history of the earth should therefore be taught in college courses as a succession of complex dynamic events, great in the past and great in future potentialities.

The formations and configurations left by the successive phases of action are to be studied primarily as the vestiges of the processes that gave them birth, and hence as their historic credentials. They are to be looked upon less as the vital things in themselves, than as the record of the events of the time and as the forerunners of the subsequent events that may be potential in them. And so, primarily, the geologic records are to be scrutinized to find the deeper meanings which they embody, whether such meanings lie in the physical, the biological, or the psychological world.

Geology the means of developing scientific imagination of time and space

Turning to specific phases of the subject, it may first be noted that geology is singularly suited to develop clear visions of vast stretches of time; it opens broad visions of the panorama of world events, a panorama still passing before us. While the celestial order of things no doubt involves greater lapses of time, these are not so easily realized, for they are not so well filled in with a succession of records of the passing stages that make up the whole. But even the lapses of geologic time are greater than immature minds can readily grasp; however, their powers of realization are greatly strengthened by studying so protracted a record, built up stage upon stage. The very slowness with which the geologic record was made, as well as the evidences of slowness in each part of the record, help to draw out an appreciation of the immensity of the whole. The round period covered by the more legible range of the geologic record rises to the order of a hundred million years, perhaps to several hundred million years. The large view of history which this implies has already come to form the ample background on which are projected the concepts of the broader class of thinkers; such largeness of view will quite surely be held to be an indispensable prerequisite to the still broader thinking of the future for which the better order of students are now preparing.

While this is preeminently true of the concept of time, the concept of space is fairly well cultivated by geologic study, though far less effectively than is done by astronomical study. Astronomy and geology work happily together in contributing to largeness of thought.

The study of the origin and early history of the earth brings the student into touch with the most far-reaching problems that have thus far called forth the intellectual efforts of man. If rightly handled, these great themes may be made to teach the true method of inquiry into past natural events whose vastness puts them quite beyond the resources of the laboratory. This method finds its key in a search for the history of such vast and remote events by a scrutiny of the vestiges these events have left as their own automatic record. This method stands in sharp contradistinction to simple speculation without such search for talismanic vestiges, a discredited method which is too often supposed to be the only way of dealing with such themes. To be really competent in the field of larger and deeper thinking, every courageous mind should be able to cross the threshold of any of the profound problems of the universe with safe and circumspect steps, however certain it may be that only a slight measure of penetration of the problem may be attainable. A well-ordered mind will remain at once complacent and wholesome when brought to the limit of its effort by the limit of evidence. The problem of the origin of celestial worlds, of which the genesis of the earth is the theme of largest human interest, is admirably suited to give college students at once a modest sense of their limitations and a wholesome attitude toward problems of the vaster type. Without having acquired the power to make prudent and duly controlled excursions into the vaster fields of thought, the mind can scarcely be said to have been liberalized.

Geology a means of training in thinking in scientific experiences

From the very outset, the tracing of the earth history forces a comprehensive study of the co-workings of the three dominant states of matter massively embodied in the atmosphere, the hydrosphere, and the lithosphere, the great terrestrial triumvirate. The strata of the earth are the joint products of these three elements and constitute their lithographic record. These three cooperating and contending elements not only bring into view the three typical phases of physical action, but they present this action in such titanic aspects as to force the young mind to think along large lines, with the great advantage that these actions are controlled by determinate laws, while the causes and the results are both tangible and impressive.

While there is a large class of tangible and determinate problems of this kind, embracing shiftings of matter on the earth's surface, distortions of strata, and changes of bodily form, there are also problems of a more hidden nature such as internal mutations. These give rise to mathematical, physical, and chemical inquiries while at the same time they call into play the use of the scientific imagination and are thus rich in the possibilities of training. Thus in varied ways geological work joins hands with chemical, physical, mechanical, and mathematical work.

When life first appears in the record, there is occasion to raise the profound question of its origin, and with this arises a closely related question as to the nature of the conditions that invited life, which leads on to the further question, what fostered the development of life throughout its long history? While the obscurity of the earliest record leaves the question of origin indeterminate for the present, duly guarded thought upon the subject should foster a wholesome spirit toward inquiry in this vital line as well as a hospitable attitude toward whatever solution may finally await us. In all such studies the student should be invited to look to the vestiges left automatically by the process itself for the answer, and he should learn to accept the teachings of evidence precisely as it presents itself. So also when a problem is, for the present, indeterminate, it is peculiarly wholesome for the inquirer to learn to rest the case where the light of evidence fails, and to be complacent in such suspension of judgment and to wait further light patiently in serene confidence that the vestiges left by the actuating agencies in their constructive processes are the surest index of the ultimate truth and are likely to be sooner or later detected and read truly.

Relation of geology to botany, zoology, psychology, and sociology

In the successive records of past life impressed on strata piled one upon another until they form the great paleontologic register, there is an ample and a solid basis for the study of the historic evolution of life. With this also go evidences of the conditions that attended this life progress and that gave trend to it. This record of the relations of life to the environing physical conditions forms one of the most stimulating fields of study that can engage the student who seeks light on the great problems of biological progress. Here geology joins hands with botany and zoology in a mutual helpfulness that is scarcely less than indispensable to each.

Following, or perhaps immediately attending, the introduction of physiological life, there appeared signs of sentient life. The preservation of certain of the sense organs, taken together with the collateral evidences of sense action, as early as Cambrian times, furnish the groundwork for a historical study of the progress of sentient life, eventuating in the higher forms of mental life. Here the problems of geology run hand in hand with the problems of psychology. The limitations of the evidence bearing on psychological phenomena, while regrettable, are not without some compensation in that they center the attention on the simpler aspects of the protracted deployment of the psychological functions.

In addition to the clear evidences of psychic action, in at least its elementary forms, there appeared early in the stratigraphic records intimations of some of the relationships that sentient beings then bore to one another; and this relationship gives occasion to study the primitive aspects of sociological phenomena. If nothing more is learned than the important lesson that sociology is not a thing of today, not an untried realm inviting all kinds of ill-digested projects, but on the contrary is a field of vast and instructive history, the gain will not be inconsiderable. There are intimations of the early existence and effective activity of those affections that precede and that cluster about the parental relationship, the nucleus of the most vital of all the sociological relationships. In contrast to the affections, there are distinct evidences of antagonistic relations, of pursuit and capture, of attack and defense; there were tools of warfare and devices for protection. In time, a wide-ranging series of experiments, so to speak, were tried to secure advantage, to avoid suffering, to escape death, and to preserve the species. There were even suggestions of the cruder forms of government. The many stages in the evolution of the various devices, as well as the stages of their abandonment, that followed one another in the course of the ages recorded the results of a multitude of efforts at sociological adjustment. They raise the question whether a common set of guiding principles does not underlie all such relationships, earlier and later, whatever their rank in our scale of valuation. And so this great field of inquiry—too narrowly regarded as merely humanistic—comes into view early in the history of the earth. The geological and the sociological sciences find in it common working ground. If the geologic and the humanistic sciences are given each their widest interpretation and their freest application, the advantage cannot be other than mutual.

It is perhaps not too much to say that studies in the physiological, the psychological, the sociological, and the allied fields necessarily lack completeness if they do not bring into their purview the data of their common historical record traced as far back as it is found to contain intimations of their actual extension.

It is customary to speak of the geologic ages as though they were wholly past; they are, indeed, chiefly past as the record now stands, but time runs on and earth history continues; the processes of the past are still active, and they are likely to work on far into the future. And so geologic study links itself fundamentally into all such present terrestrial interests as take hold of the distant future. The forecast of the earth's endurance, attended by conditions congenial to life and to the mental and moral activities, hinges on a sound insight into the great actuating forces inherent in the earth, together with those likely to come into play from the celestial environment. All human interests, in so far as they are dependent on a protracted future, center in the prognosis of the earth based on its present and its past. The latest phases of geologic doctrine prophesy a long future habitability of the earth. They thus give meaning and emphasis to the deeper purposes sought in all the higher endeavors, not the least of which is education, particularly those phases of education that lead to effects which may be handed down from age to age.

Standard for selecting subject matter for the general college course: select fundamentals or that which bears on fundamentals

Out of all this vast physical, biological, and psychological history, the things to be selected for substance of thought and for service in mental training in a college course are, first of all, those that are either fundamental in themselves, or that have vital bearings on what is fundamental. These are chiefly the great dynamic factors, the agencies that gave trend to the master events, the forces that actuated the basal processes by which the vast results were attained. The material formations and the surficial configurations that resulted are to be duly considered, to be sure, for they form the basis of interpretation and they are, besides, the repositories of economic values of indispensable worth; but, as already urged, in a course of intellectual training, these are to be regarded rather as the relics of the great agencies and the proofs of their actions, than as the most vital subjects of study, which are the agencies themselves. As already remarked, the geologic formations are to be treated rather as the credentials of the potencies that reside in the earth organism, than as the vital things themselves. The vestiges of creation and the footprints of historical progress embody the soul of the subject; they constitute the chief source of inspiration to those who aspire to think in large, deep ways of really great things. It is of little value, from the viewpoint of liberal culture, to know that there is a certain succession of sandstones, shales, and limestones; that professional convention has given them certain names, more or less infelicitous in derivation and in phonic quality; but it is of vital consequence to learn how and why these relics of former processes came to be left as they were left, and thus came to be witnesses to the history of the far past. It was a wise thing, no doubt, that the fathers of geology strongly insisted that there should be a rigorous and rather literal adhesion to the terrestrial record in all earth studies, because in those times of transition from the loose, more or less fantastic thought that marked the adolescent stage of the human race, it was imperative that students should stick close to the immediate evidence of what had transpired, and should withhold themselves from much enlargement of view based on the less tangible evidences; but at the present stage, when the general nature of the earth's history has been firmly established, it would be an error on the part of those who seek for the most liberalizing and broadening values of the science, to treat the record merely as a material register of immediate import only, to the neglect of the less tangible but more vital teachings immanent in its great forces and processes. The seeker of liberal culture should direct his attention to the great events, and, above all, to the larger and deeper meanings implied by these events.

And so—may I be pardoned for reemphasizing?—the teacher of geology whose essential purpose is liberal training, leading to broad and firm knowledge and to sound processes of thought, will critically observe the distinction between geology taught appropriately from the collegiate point of view, and geology taught specifically from the professional and technical points of view. In these latter, specific details in specific lines are important, and may even be essential, but it is the function of the college teacher of geology to select from the great mass of material of the science such factors as are basal, vital, and talismanic. He will give these emphasis, while he neglects the multitude of details that lack significance as working elements or as landmarks of progress, whatever their value in other relations. This selection is equally important, whether applied to the great physical processes that have shaped the earth into its present configuration, or to the great chemical and mineralogical processes that have determined its texture and its structure, or to the great biological and psychological processes that have given trend to the development of its inhabitants.

Even if the undergraduate course in geology is pursued less for the purpose of liberal culture than as a means of preparing for a professional career as an economic geologist, no essential departure from an effort to master first the basal features and the broader aspects of the science, especially the dynamic aspects, is to be advised. The shortest road to declared success in professional and economic geology lies through the early mastery of its fundamentals. No doubt immediate and apparent success may often be sooner reached by a narrower and shallower study of such special phases of the subject as happen just now to be most obviously related to the existing state of the industries; but industrial demands are constantly changing—indeed, at present, rather rapidly—and new aspects follow one another in close succession. These new aspects almost inevitably spring from the more basal factors as these rise into function with the progress of experience or the stress of new demands. Those who have sought only the immediate and the superficial, at the expense of the basal, and especially those who have neglected to acquire the power and the disposition to search out the fundamentals, are quite sure to be left among the unfortunates who trail behind; they are little likely to be found among those who lead at the times when leadership counts. In the judgment of those master minds that lead in affairs and that take large and penetrating views, the lines along which the most vital contributions to economic interests are being made connect closely with basal studies of the actuating agencies that condition great enterprises. In the judgment of the writer, it is a false view to suppose that any short, superficial study of so vast a subject as the constitution and history of the earth can result in economic competency. In so far as time for study is limited, it should be concentrated on the great underlying factors that constitute the essentials of the science. It is here assumed that men who care to take a college course at all are seeking for a large success and are ambitious for a high personal career. If they look ultimately to professional work in economic lines, they may safely be advised that the straight road to declared success lies in a search for the vital forces, the critical agencies, and the profound principles that make for great results, not along the by-paths whose winding, superficial courses are turned hither and thither by adventitious conditions whose very nature invites distrust rather than confidence.

Evaluations of methods of teaching

Turning to some of the more formal phases of treatment, three types of work are presented: (1) the use of nature's laboratory, the world itself, (2) the use of the college collections and laboratories, and (3) the use of the literature of the subject.

(1) Fortunately, there is no place on the face of the earth where there is not some natural material for geologic study, for even in the most artificialized locations geological processes are active. In crowded cities these processes may be easily overlooked, but yet they are susceptible of effective use. Within easy access from almost every college site there are serviceable fields of study, and these, in any live course, will be assiduously cultivated. They may be relatively modest in their phenomena; they may seem to lack that impressiveness which has played so large a part in the popular notion of the content of geology, but they may nevertheless serve as most excellent training grounds for young geologists. If students are so situated as to be brought at the beginning of study under the influence of very impressive displays of geologic phenomena—precipitous mountains, rugged cliffs, deep canyons, and the like—there is danger that their mental habits may become diffusive rather than close and keen; the emotions may be called forth in wonder rather than turned into zest in the search for evidence. If students are to be trained to diligence in inquiry and to the highest virility in inference and interpretation, it is perhaps fortunate for them if they are located where only modest records of geological processes are presented for study. In such regions they are more likely to be led to scrutinize the field keenly, sharply, and diligently for data on which to build their interpretations. The scientific use of their imaginations is all the better trained if, in their endeavor to build up a consistent concept of the whole structure that underlies their field, they are forced to project their inferences from a few out-crops far beneath the cover of the adjacent mantle that shuts off direct vision. Few teachers have, therefore, any real occasion to long for richer fields than those accessible to them, if they have the tact to render these fertile in stimulus and suggestion.

(2) Laboratory work upon the material collected in the field work, as well as laboratory work upon the college collections, are essential adjuncts. Ample provisions for this supplementary work, however modest the appointments, are important and can usually be secured by ingenuity and diligence in spite of financial limitations.

Both field and laboratory work should be well correlated with one another and with the systematic work on the text that guides the study, so that each shall whet the edge of the other and all together accomplish what neither could alone.

(3) The text selected should be such as lends itself, in some notable degree at least, to the general purposes set forth above. It should be supplemented, so far as may be, by judicious assignments for reading and for special study. Lectures may be made a valuable aid to the discussions of the classroom, but with college classes they can rarely be made an advantageous substitute for the discussions. Lecturing, so far as used, is best woven informally into the classroom discussions. Supplementary lecturettes may be advised if they are of such an informal sort that they may almost unconsciously take their start from any vital point encountered in the course of discussion, may run on as far as the occasion invites, and may then give way again to the discussion with the utmost informality. Such little participations in the work of the classroom, on the part of the teacher, are likely to be cordially welcomed. At the same time, if well done, they will set an excellent example in the presentative art as also in an apt organization of thought.

Organization of courses

If the stated course in earth-science is limited to the junior and senior year by the existing requirements of the curriculum of the institution or by the rulings of its officers—as is not uncommonly the case at present—it is relatively immaterial whether the sections of the course are marshaled under the single name "geology" or whether they are given separate titles as sub-sciences, provided the special subjects are arranged in logical sequence and in consecutive order. If, on the other hand, the teacher's choice of time and relations is freer, the more accessible phases of earth study, now well organized under the name of "physiography," form an excellent course for either freshmen or sophomores. It opens their minds to a world of interesting activities about them which have probably been largely overlooked in previous years. It gives them substance of thought that will be of much service in the pursuit of other sciences. It has been found that it is not without rather notable service to young students as the basis of efforts in the art of literary presentation, a felicity to which teachers of this important art frequently give emphatic testimony. The secret seems to lie in the fact that physiography gives varied and vivid material susceptible of literary presentation, while the fixed qualities of the subject matter control the choice of terms and the mode of expression.

If geography and physiography are given in the earlier years, the course in historical geology, as well as the study of the more difficult phases of geological processes, of the principles of dynamic geology, together with mineralogy, petrology, and paleontology, may best fall into the later years, even if some interval separates them from the geography and physiography.

One hundred and twenty classroom hours, or their equivalent in laboratory and field work, are perhaps to be regarded as the irreducible minimum in a well-balanced undergraduate course, while twice that time or more is required to give a notably strong college course in earth-science.

A consideration of the sequences among the geological sub-subjects, as also among the subjects that are held to be preliminary to the earth-sciences, is important, but it would lead us too far into details which depend more or less on local conditions. In the experience of American teachers it appears to have been found advisable to put geological processes and typical phenomena to the front and to take up geological history afterwards. The earlier method of taking up the history first, beginning with recent stages and working backward down the ages,—once in vogue abroad,—has been abandoned in this country. It was the order in which the science was developed and it had the advantage of starting with the living present and with the most accessible formations, but this latter advantage is secured by studying the living processes, as such, first, and turning to the history later. This permits the study of the history in its natural order, which seems better to call forth the relations of cause and effect and to give emphasis to the influence of inherited conditions.

Respecting antecedents to the study, the more knowledge of physics, chemistry, zoology, and botany, the better, but it is easy to over-stress the necessity for such preparation, however logical it may seem, for in reality all the natural sciences are so interwoven that, in strict logic, a complete knowledge of all the others should be had before any one is begun, a reductio ad absurdum. The sciences have been developed more or less contemporaneously and progressively, each helping on the others. They may be pursued much in the same way, or by alternations in which each prior study favors the sequent one. They may even be taken in a seemingly illogical order without serious disadvantage, for the alternative advantages and other considerations may outweigh the force of the logical order, which is at best only partially logical. It is of prime importance to stimulate in students a habit of observing natural phenomena at an early age. It may be wise for a student to take up physiography, or its equivalent, early in the college course, irrespective of an ideal preparation in the related sciences. It is unfortunate to defer such study to a stage when the student's natural aptitude for observation and inference has become dulled by neglect or by confinement to subjects devoid of naturalistic stimulus. To permit students to take up earth-science in the freshman and sophomore years, even without the ideal preparation, is therefore probably wiser than to defer the study beyond the age of responsiveness to the touch of the natural environment. The geographic and geologic environment conditioned the mental evolution of the race. It left an inherited impress on the perceptive and emotional nature, only to be awakened most felicitously, it would seem, at about the age at which the naturalistic phases of the youth's mentality were originally called into their most intense exercise.

T. C. CHAMBERLIN The University of Chicago



VIII

THE TEACHING OF MATHEMATICS

Recent changes and some of their sources

In recent years the teaching of mathematics has undergone remarkable changes in many countries, both as regards method and as regards content. With respect to college mathematics these changes have been evidenced by a growing emphasis on applications and on the historic setting of the various questions. To understand one direct source of these changes it is only necessary to recall the fact that in about 1880 there began a steady stream of American mathematical students to Europe, especially to Germany. Most of these students entered the faculties of our colleges and universities on their return to America It is therefore of great importance to inquire what mathematical situation served to inspire these students.

The German mathematical developments of the greater part of the nineteenth century exhibited a growing tendency to disregard applications. It was not until about 1890 that a strong movement was inaugurated to lay more stress on applied mathematics in Germany.[3] Our early American students therefore brought with them from Germany a decided tendency toward investigations in mathematical fields remote from direct contact with applications to other scientific subjects, such as physics and astronomy, which had so largely dominated mathematical investigations in earlier years.

This picture would, however, be very incomplete without exhibiting another factor of a similar type working in our own midst. J. J. Sylvester was selected as the first professor of mathematics at Johns Hopkins University, which opened its doors in 1876 and began at once to wield a powerful influence in starting young men in higher research. Sylvester's own investigations related mainly to the formal and abstract side of mathematics. Moreover, "he was a poor teacher with an imperfect knowledge of mathematical literature. He possessed, however, an extraordinary personality; and had in remarkable degree the gift of imparting enthusiasm, a quality of no small value in pioneer days such as these were with us."[4]

Influence of researches in mathematics on methods of teaching

Mathematical research was practically introduced into the American colleges during the last quarter of the nineteenth century, and the wave of enthusiasm which attended this introduction was unfortunately not sufficiently tempered by emphasis on good teaching and breadth of knowledge, especially as regards applications. In fact, the leading mathematician in America during the early part of this period was glaringly weak along these lines. By means of his bountiful enthusiasm he was able to do a large amount of good for the selected band of gifted students who attended his lectures, but some of these were not so fortunate in securing the type of students who are helped more by the direct enthusiasm of their teacher than by the indirect enthusiasm resulting from good teaching.

The need of good mathematical teaching in our colleges and universities began to become more pronounced at about the time that the wave of research enthusiasm set in, as a result of the growing emphasis on technical education which exhibited itself most emphatically in the development of the schools of engineering. While the student who is specially interested in mathematics may be willing to get along with a teacher whose enthusiasm for the new and general leads him to neglect to emphasize essential details in the presentation, the average engineering student insists on clearness in presentation and usability of the results. As the latter student does not expect to become a mathematical specialist, he is naturally much more interested in good teaching than in the mathematical reputation of his teacher, even if his reputation is not an entirely insignificant factor for him.

During the last decade of the nineteenth century and the first decade of the present century the mathematical departments of our colleges and universities faced an unusually serious situation as a result of the conditions just noted. The new wave of research enthusiasm was still in its youthful vigor and in its youthful mood of inconsiderateness as regards some of the most important factors. On the other hand, many of the departments of engineering had become strong and were therefore able to secure the type of teaching suited to their needs. In a number of institutions this led to the breaking up of the mathematical department into two or more separate departments aiming to meet special needs.

In view of the fact that the mathematical needs of these various classes of students have so much in common, leading mathematicians viewed with much concern this tendency to disrupt many of the stronger departments. Hence the question of good teaching forced itself rapidly to the front. It was commonly recognized that the students of pure mathematics profit by a study of various applications of the theories under consideration, and that the students who expect to work along special technical lines gain by getting broad and comprehensive views of the fundamental mathematical questions involved. Moreover, it was also recognized that the investigational work of the instructors would gain by the broader scholarship secured through greater emphasis on applications and the historic setting of the various problems under consideration.

To these fundamental elements relating to the improvement of college teaching there should perhaps be added one arising from the recognition of the fact that the number of men possessing excellent mathematical research ability was much smaller than the number of positions in the mathematical departments of our colleges and universities. The publication of inferior research results is of questionable value. On the other hand, many who could have done excellent work as teachers by devoting most of their energies to this work became partial failures both as teachers and as investigators through their ambition to excel in the latter direction.

Range of subjects and preparation of students

It should be emphasized that the college and university teachers of mathematics have to deal with a wide range of subjects and conditions, especially where graduate work is carried on. Advanced graduate students have needs which differ widely from those of the freshmen who aim to become engineers. This wide range of conditions calls for unusual adaptability on the part of the college and university teacher. This range is much wider than that which confronts the teachers in the high school, and the lack of sufficient adaptability on the part of some of the college teachers is probably responsible for the common impression that some of the poorest mathematical teaching is done in the colleges. It is doubtless equally true that some of the very best mathematical teaching is to be found in these institutions.

In some of the colleges there has been a tendency to diminish the individual range of mathematical teaching by explicitly separating the undergraduate work and the more advanced work. For instance, in Johns Hopkins University, L. S. Hulburt was appointed "Professor of Collegiate Mathematics" in 1897, with the understanding that he should devote himself to the interests of the undergraduates. In many of the larger universities the younger members of the department usually teach only undergraduate courses, while some of the older members devote either all or most of their time to the advanced work; but there is no uniformity in this direction, and the present conditions are often unsatisfactory.

The undergraduate courses in mathematics in the American colleges and universities differ considerably. The normal beginning courses now presuppose a year of geometry and a year and a half of algebra in addition to the elementary courses in arithmetic, but much higher requirements are sometimes imposed, especially for engineering courses. In recent years several of the largest universities have reduced the minimum admission requirement in algebra to one year's work, but students entering with this minimum preparation are sometimes not allowed to proceed with the regular mathematical classes in the university.

Variety of college courses in mathematics

Freshmen courses in mathematics differ widely, but the most common subjects are advanced algebra, plane trigonometry, and solid geometry. The most common subjects of a somewhat more advanced type are plane analytic geometry, differential and integral calculus, and spherical trigonometry. Beyond these courses there is much less uniformity, especially in those institutions which aim to complete a well-rounded undergraduate mathematical course rather than to prepare for graduate work. Among the most common subjects beyond those already named are differential equations, theory of equations, solid analytic geometry, and mechanics.

A very important element affecting the mathematical courses in recent years is the rapid improvement in the training of our teachers in the secondary schools. This has led to the rapid introduction of courses which aim to lead up to broad views in regard to the fundamental subjects. In particular, courses relating to the historical development of concepts involved therein are receiving more and more attention. Indirect historical sources have become much more plentiful in recent years through the publication of various translations of ancient works and through the publication of extensive historical notes in the Encyclopedie des Sciences Mathematiques and in other less extensive works of reference.

The problem presented by those who are preparing to teach mathematics may at first appear to differ widely from that presented by those who expect to become engineers. The latter are mostly interested in obtaining from their mathematical courses a powerful equipment for doing things, while the former take more interest in those developments which illumine and clarify the elements of their subject. Hence the prospective teacher and the prospective engineer might appear to have conflicting mathematical interests. As a matter of fact, these interests are not conflicting. The prospective teacher is greatly benefited by the emphasis on the serviceableness of mathematics, and the prospective engineer finds that the generality and clarity of view sought by the prospective teacher is equally helpful to him in dealing with new applications. Hence these two classes of students can well afford to pursue many of the early mathematical courses together, while the finishing courses should usually be different.

The rapidly growing interest in statistical methods and in insurance, pensions, and investments has naturally directed special attention to the underlying mathematical theories, especially to the theory of probability. Some institutions have organized special mathematical courses relating to these subjects and have thus extended still further the range of undergraduate subjects covered by the mathematical departments. The rapidly growing emphasis on college education specially adapted to the needs of the prospective business man has recently led to a greater emphasis on some of these subjects in several institutions.

The range of mathematical subjects suited for graduate students is unlimited, but it is commonly assumed to be desirable that the graduate student should pursue at least one general course in each one of broader subjects such as the theory of numbers, higher algebra, theory of functions, and projective geometry, before he begins to specialize along a particular line. It is usually taken for granted that the undergraduate courses in mathematics should not presuppose a knowledge of any language besides English, but graduate work in this subject cannot be successfully pursued in many cases without a reading knowledge of the three other great mathematical languages; viz., French, German, and Italian. Hence the study of graduate mathematics necessarily presupposes some linguistic training in addition to an acquaintance with the elements of fundamental mathematical subjects.

Historical studies make especially large linguistic demands in case these studies are not largely restricted to predigested material. This is particularly true as regards the older historical material. In the study of contemporary mathematical history the linguistic prerequisites are about the same as those relating to the study of other modern mathematical subjects. With the rapid spread of mathematical research activity during recent years there has come a growing need of more extensive linguistic attainments on the part of those mathematicians who strive to keep in touch with progress along various lines. For instance, a thriving Spanish national mathematical society was organized in 1911 at Madrid, Spain, and in March, 1916, a new mathematical journal entitled Revista de Matematicas was started at Buenos Aires, Argentine Republic. Hence a knowledge of Spanish is becoming more useful to the mathematical student. Similar activities have recently been inaugurated in other countries.

History of college mathematics

Until about the beginning of the nineteenth century the courses in college mathematics did not usually presuppose a mathematical foundation carefully prepared for a superstructure. According to M. Gebhardt, the function of teaching elementary mathematics in Germany was assumed by the gymnasiums during the years from 1810 to 1830.[5] Before this time the German universities usually gave instruction in the most elementary mathematical subjects. In our own country, Yale University instituted a mathematical entrance requirement under the title of arithmetic as early as 1745, but at Harvard University no mathematics was required for admission before 1803.