ELEMENTARY MATHEMATICS IN COLLEGES

COLLEGE MATHEMATICS 747

ELEMENTARY MATHEMATICS IN COLLEGES.By DR. WILLIAM E. ROTH,

Assistant Professor of Mathematics, University of Wiscon-sin, Extension Division, Milwaukee, Wis.

Mathematicians are prone to disregard the utility of thesubjects they like most to study and to teach, moreover theydo not like to justify their pet subject on practical grounds.The students in colleges, on the other hand, want somedefinite knowledge of the utility of courses they are obligedto take; likewise educators upon whom the responsibilityof preparing young people for life rests must feel convincedthat a knowledge of required courses has either a culturalor a practical value in life. For mathematics we shallhere emphasize the utilitarian value, though we do notdoubt that a subject which touches so many fields of thoughthas likewise great cultural value.

Mathematics is a powerful tool for the research man inany science and for the engineer; it is as much a meansof understanding scientific and technical subjects as it isan aid in applying the knowledge of them. Therefore, wesometimes find successful men in these fields who rarelyuse the advanced mathematics which they had learned incollege, but who in acquiring their knowledge needed it tomaster the principles they must constantly apply in theirwork. Without calculus it is impossible to study and tounderstand dynamics�the science of motion. Thus thestudent will meet with calculus in hydrodynamics, electro-dynamics, aerodynamics, in the dynamics of the electronand of the solar system. Calculus is the mathematics ofmotion; in fact, it was through his studies of effects offorces in a moving system that Newton was led to discover,or if you please, to create this eminently potent branchof mathematics. But calculus is equally effective in thestudy of statics, that is, of forces in equilibrium or at rest.Thus torques and strains of beams, bending moments,moments of inertia, pressure of fluids, work, power, poten-tial, the curve of and the forces acting upon the cable sus-taining a suspension bridge and the pressure exerted uponmembers of an arch may be computed from appropriatealgebraic formulae that are solutions of surprisingly simpledifferential equations. The prospector now relies on elec-trical, magnetic, and seismographie methods to aid him in

748 SCHOOL SCIENCE AND MATHEMATICS

locating valuable mineral deposits and these methods arebased on mathematical considerations that can be under-stood only when one is conversant with calculus. The itemswe have mentioned, though the list is not comprehensive,touch nearly every branch of engineering and are there-fore indicative of the importance of calculus in that field.The examples of the applications of mathematics that

we have cited are taken almost exclusively from the domainof physics so that we have already indicated the significantrole of calculus in this study. Modern physical researchhas long out-stripped the bare mathematical needs of theengineer. Indeed physicists are crowding mathematiciansin their eagerness to grasp new mathematical knowledgeand are themselves making mathematical discoveries,hoping by means of this new knowledge to discover newfacts about the physical world. The powerful aid ofmathematics to physics has been indicated above only inits most elementary aspects; in regard to its more pro-found applications in that field A. W. Stern says: "Thetendency of this new physics is to operate with mathemati-cal symbols and not with physical ideas and concepts de-rived from sense experience.9fl That this will probably bethe case is further emphasized by the chemist, Irving Lang-muir, who, in giving the presidential address before theAmerican Chemical Society (Sept. 1929) said: "We haveno guarantee whatever that nature is so constructed thatit can’ be adequately described in terms of mechanical orelectrical models; it is much more probable that our mostfundamental relationships can only be expressed mathe-matically if at all.9’2 Whether or not these opinions will beborne out by the experience of time, it is nevertheless prob-ably true that the modern physical theories, however irra-tional they may appear, cannot be broken down nor im-proved by one who does not understand their mathemati-cal foundations; therefore he, who wishes to aid in theaccomplishment of such ends, cannot follow a wiser coursethan to prepare himself in mathematics.Now, as is well known, chemistry studies the changes of

matter and so likewise does biology; hence calculus, the

^tern. A. W., The Role of Mathematics in Modern Physical Theory. Monist., Vol.39 (April, 1929), pp. 263-272.ângmuir, Irvinp, Modern Concepts in Physics and Their Relation to Chemistry,

Journal of the American Chemical Society. Vol. 61 (1929). pp. 2847-2868.


mathematics of motion/should aid in the study and in theinvestigation of these subjects. To show that this is reallythe case we shall quote men whose interest is not that ofteaching and studying mathematics but is that of further-ing the advance of these sciences. Thus Farrington Danielssays: "Chemistry has graduated from the class of descrip-tive sciences into the class of exact sciences and has takenits place by the side of physics and engineering as a branchof mathematics. All chemical phenomena and many biologi-cal phenomena are now being interpreted in terms of physi-cal chemistry, and physical chemistry cannot be masteredwithout a foundation in mathematics which extends throughcalculus. Research chemists are keen to realize their de-pendence on mathematics and many of them feel blockedin their progress by their weakness in mathematics."3 H.L. Adams in his address upon retiring from the Presidencyof the Washington Chemical Society said: "Actually theamount of pure "mathematics required in most branches ofchemistry is small. Elementary calculus is as far as mostneed go�.ff and again, "The internal structure of the atomis no longer a complete mystery. The physicists have ap-parently claimed this territory for their own and it mustbe admitted that the great fundamental discoveries in thisfield have been made by^ physicists. Their remarkableprogress has been largely due to the fact that they did notfail to use all possible mathematical tools including themost profound analytical methods."* That the study ofatomic structure might be taken out of the chemist’s handswas foreseen by A. Crum Brown for we find that nearlyforty years ago he made the prediction: ."But unless he(the young chemist) learns the langzmge of the empire (ofmathematics), he will become a provincial, and the higherbranches of chemical work, those which require reason aswell as skill, will gradually pass out of his Jiands. Let themwhile there is time learn the language of the empire. Letthem become fluent and ready in its use; let them read withcare the work that is being done on the border betweenchemistry and mathematical physics, and, as they find op-portunity, do such work themselves, and so be ready to take

Daniels, Farrington, Mathematics for Students in Chemistry, American Mathe-matical Mwithly. Vol. 35 (1928). pp. 3-9.Âdams, H., Chemistry As a Branch of Mathematics, Journal of the Washington

Academy of Sciences, Vol. 16 (1926). pp. 266-276.


their part in the union which will certainly come."5 Theoutcome as Professor Brown predicted it has in a measurecome to pass; fortune smiled on the physicists, partly be-cause they were quick to accept mathematics as an aid intheir research and partly because the discovery of X-raysand radio-activity, which initiated the recent advances inphysics, were regarded as physical and not chemical phe-nomena. The shortcomings in the past are being rapidlycorrected, chemists are beginning to speak fluently the lan-guage of mathematics and no one who wishes to enter theirfundamental field can afford to neglect learning the lan-guage in which its affairs are so largely conducted.

An interesting article by 0. W. Richard on "The Mathe-matics of Biology" cites several very significant applicationsthat have been found for mathematics in that science. Wequote the following excerpts from Richard’s paper:"�anyone who even idly turns the pages of the Journal ofGeneral Physiology must be forcibly reminded of his cal-culus text. Until recently one could read by omitting themathematics. Noiv, however, it is necessary to follozv themathematics in order to understand the logic because thisform of expression permits the presentation of such infor-mation in a more concise manner." Again he says, "Withthe present increase of mathematical usage in experimentalliterature even the medical student must be familiar withthe more general facts of calculus and biometrics.9^ Andmore recently Harris said, "The tendency of the times isunmistakable; the demand for quantitative work is moreand more dominant in the biology of today," and "That aman should be able to reason about highly complicated phe-nomena without the use of mathematical formulae is nomore remarkable than that he should be unable to see chro-mosomes without the microscope.9" The quotations, takenas they are, from the writings of men who are not primarilyinterested in mathematics, are very significant as an indi-cation of the esteem in which scientists hold the subject.

^rown, A. Crum, Presidential Address, Transaction of the Chemical Society (Lon-don). Vol. 61 (1892), p. 478.

Richards.- 0. W., The Mathematics of Biology, American Mathematical Monthly,Vol. 32 (1925), pp. 30-36.

^Harris, J. A., Mathematics in Biology, �The Scientific Monthly (August, 1929),pp. 141-152.


The frequency with which we are obliged to refer to cal-culus and the importance of the connections in which thereferences occurred must cause us to agree with ThomasHill, a former president of Harvard University when hesaid that: "The discoveries of Newton have done more forEngland and for the race than has been done by wholedynasties of British monarchs.9^ For Newton’s. greatestdiscovery and the one that made the others possible wascalculus.The exposition of the utility of mathematics that has been

presented thus far is concerned primarily with calculus.Consequently the reasons that algebra, trigonometry, andanalytic geometry ought to be required of college studentsmust still be discussed. One of the principal reasons forthe requirement is that these courses are the stepping stonesto calculus. Newton and Leibnitz could not have discovered,or, as some prefer to say, created calculus without theanalytic geometry of Descartes, for its logic is too subtleto be appreciated without the aid of the pictures that thisgeometry gives, and we now cannot master the subjectwithout the same aid. Again analytic geometry is basedon algebra and trigonometry. Hence, these three subjects,algebra, trigonometry, and analytic geometry, are cer-tainly useful then for those who will need calculus in theirprofession.

If, however, one is not interested in the sciences nor inthe engineering profession, he should, in order that he maybecome an intelligent and learned man, gain a substantialknowledge of mathematics. Thinkers in all fields of humanendeavor and progress must concern themselves with itspure logic and with its universal utility. "Who does notknow mathematics and the results of recent scientific inves-tigation dies without knowing truth/" And the philosopherHegel said, "Whoever does not know the works of the an-cients has lived without knowing beazity^ But in thesciences only the mathematics and the philosophy, if phi-losophy is a science, of the ancients are entirely valid today.Their mathematics is now studied as arithmetic, algebra,and geometry. It is a simple matter to find many words of

îll, Thomas, The Imagination in Mathematics, North American Keview, Vol. 85(1857), pp. 223-237."Shellbach, "Hegel, quoted by J. W. A. Young, Teaching of Mathematics (N. Y..

1907), p. 44.


praise of mathematics as cultural background though somerefute such a claim for the subject. Florian Cajori, inMathematics in Liberal Education^ has given a rather com-plete collection of opinions on this aspect of the subject. Acareful perusal of this book indicates that the followingopinion as expressed by Biertholot will summarize the feel-ings that mathematics has quite generally inspired:"Mathematics gives the young man a clear idea of demon-stration and habituates him to form long trains of thoughtand reasoning methodically connected and sustained by afinal certainty of results; and it has the further advantage,from a purely moral point of mew, of inspiring an absoluteand fanatical respect for the truth."12

There is now growing up a new appreciation of the cul-tural value of mathematics which arises out of recent devel-opments in philosophy. This new attitude finds expressionin the works of Bertrand Russell, A. N. Whitehead, Cas-sius Keyser, and of the Dutch philosopher Brouwer; its out-come and the part mathematical thinking will play in phi-losophy is. a matter of conjecture for the present. Theseidealized statements cover muted ground and consequentlyare convincing only to him who agrees with them. Weshall, therefore, seek to show in what respects aside frompossible cultural value, algebra, trigonometry, and analyticgeometry deserve important places in a course of study.For algebra, in itself, we can give no more utilitarian

reason for its study than we have already given; namely,that it is the main trunk supporting the higher branches ofmathematics mentioned and as the foundation upon whichthe mathematics of investments, and the mathematics ofstatistics and economics rest. These certainly leave nodoubt as to utility. To be more specific we will mentiononly the fact that the computation of sinking funds, an-nuities, depreciation, value of bonds, and of insurance areall based on formulae whose derivation requires algebraand which can be used intelligently only when their alge-braic derivations are understood. The theory of probabil-ity, a subject that is receiving more and more space in col-lege algebra textbooks, has long been very useful as a sound

"Cajori, Florian, Mathematics in Liberal Education, Christopher Publishing House,Boston (1928).

"Biertholot, M. P., Science As an Instrument in Liberal Education, Popular ScienceMonthly, Vol. 51 (1897). p. 253.


foundation for life insurance, and is now proving an indis-pensable aid in biological research and in sociology. More-over, physics and astronomy present problems in which thenumber of unknown factors makes their separate treatmentimpossible, but the probable aggregate effect of these fac-tors can be determined with a fair degree of certainty.

Trigonometry certainly is useful to the surveyor, to theengineer, to the navigator of the sea and of the air, and tothe men in some other walks of life. For the man of sciencethe most interesting and useful contribution of trigonom-etry is the trigonometric functions, particularly the sineand the cosine of an angle. One sees these functions in theripples on the surface of a pond; one hears them in music;and by means of them the radio is enabled to bring us enter-tainment and knowledge; and it is they that play a veryimportant role in calculus and its applications. Analyticgeometry is of little use in itself and becomes useful onlyas an approach to other branches of mathematics, particu-larly to calculus, and can have very fundamental utility inlife only for him who wishes to pursue mathematics further.We have said nothing regarding mathematics as a field

of research. Work here has, in the past, been carried onlargely by teachers in colleges and universities and it isdesirable that this should continue to be the case in thefuture. However, large industrial enterprises are begin-ning to feel the need of men trained in applied mathematicswho can aid in industrial research. The number of suchpositions will probably increase and the investigations thatare carried on by men holding them will gradually grow inimportance. Nevertheless, research work in mathematicswhich may have no immediate use in the arts must be car-ried on by teachers in colleges and universities. Whetherit be as a teacher or as a technician that one takes up thestudy of advanced mathematics, he will find an immensefield of research awaiting him. Very little is known whencompared with that which is still unknown. All fields ofmathematical investigation contain many buried treasuresthat await someone to uncover them. Some fields that wereonce thought to have been completely worked out have sincedisclosed much precious material and consequently the ex-istence of material that cannot now be imagined is veryprobable. We may mention two problems from the theory


of numbers to show how limited the knowledge of mathe-matics really is; and these will not be taken because oftheir importance either practically or theoretically, butbecause they can be stated in the language of arithmeticand algebra. The first is to find (by analytical methods)the next succeeding prime following the first n prime in-tegers. The second is the proof of Format’s famoustheorem: that xn+yn=’zn cannot be satisfied by integersx, y, and z, not zero, when n is an integer greater than two.This theorem has never been proved unless we accept For-mat’s statement that he has done so. When n=2 the equa-tion is satisfied by the Pythagorean numbers of which theset giving us SH^2^2 is very commonly known. Just asmany other problems have led to a host of important re-sults until finally solutions were found for them, so attemptsto solve these problems have contributed a large amountof^ mathematical knowledge to the world and. the end oftheir contribution is not yet in sight. Few problems inmathematics can be so simply stated as these, but it is nodifficult task to propose many problems which may resistattempts to solve them for many years. Theoretical mathe-matics, under which head we might classify problems likethose mentioned, must continue to pave the way for thatwhich is of immediate use. Conic sections were studiedby the Greeks many centuries before Kepler announced hisremarkable discoveries on planetary motion.

Institutions of learning should be in a position to intro-duce their capable students to a number of fields of ad-vanced scientific thought, and must provide the essentialmathematical foundations that make such an introductionpossible; otherwise they fail deplorably in the obligationsthey assume in accepting such students. There is scarcelya shadow of doubt that mathematics must play a formid-able role in the scientific thought of the future for leadersin the sciences are not only using its material in their workbut are urging their students to do likewise. Mathemati-cians, who will agree with the scientists in their opinion ofthe power and usefulness of mathematical methods, aremore cautious in its use than they, and often are led towonder if those, who so frequently use mathematicalphrases and formulae, really know what they are talkingabout. Elementary applied mathematics courses which

COLLEGE MATHEMATICS’ 755

frequently are too unmathematical to be of great value, aregiven in the departments of colleges that have need forthem because it is felt that the theoretical treatment asgiven in the regular courses of the mathematics depart-ment are unnecessarily thorough. This procedure is veryapt to lead directly to the inappropriate use of mathe--matics; a tendency against which Dr. Edwin B. Wilson13warned the members of the American Statistical Associa-tion in his presidential address only last December. Thesame warning has been expressed frequently by mathe-maticians. At the Glasgow meeting of the B. A. A. S. in1901, Professor Forsyth said: t{�that there is danger ofobtaining untrustworthy results in physical sciences, ifonly the results of mathematics are used; for the personso using the weapon can remain unacquainted with theconditions under which it can rightly be applied.’^

In order to appreciate the correct use of mathematics, theinvestigator must not only understand the derivation of thatwhich he uses but must have employed it on problems fromother than his own limited field. These conditions arerarely fulfilled by departmental courses where the point ofview is limited to a special class of problems. Calculus forthe engineer and calculus for the chemistry student cannotbe essentially more different than addition and subtractionfor the farmer are different from those employed by thebusiness man. Differentiation and integration, wîch arethe only operations studied in elementary calculus, like ad-dition and subtraction, are independent of the fields inwhich they are used. The same holds for elementary statis-tics; that used in biology is applicable in educational re-search. The inappropriate use of mathematics, due tosuperficial training such as is frequently received in de-partmental courses, is very easily remarked by one who hasa sufficient mathematical background, and will certainlyexpose those so using it to ridicule and embarrassments.Now the mathematics departments of all schools should tryto supply the needs of other departments by offering soundcourses in calculus, in the mathematics of investments and

"Wilson, Edwin B., Mathematics and Statistics, Scientific Monthly (April, 1930),pp. 294-300.

"Perry, John., Discussion on the Teaching of Mathematics, McMilIan and Co., Lon-don (reprinted, 1902).


in statistics. On the other hand, other departments mustbegin to realize that the mathematics which is essential intheir work requires more preparation and must be moreliberal in its scope than they are willing to concede. Other-wise their students and research men will more and morefall into the ridiculous situations against which Dr. Wilsonhas warned the statisticians. The danger which may arisethrough the use of more advanced mathematics will not benearly so great if the elementary courses here discussed arethoroughly understood by students.

FROM THE SCRAPBOOK OF A TEACHER OF SCIENCE.BY DUANE ROLLER,

The University of Oklahoma, Norman, Okla.

There exists no category of the sciences to which the nameof applied science could be given. We have science, and theapplications of science, which are united together as the treeand its fruit.�John Tyndall, "Lectures on Light.’9We cannot but think there is something like a fallacy

in Mr. Buckle’s theory that the advance of mankind is neces-sarily in the direction of science, and not in that of morals.�Loivell.

It is this disposition to find the real nature of the factsin the smallest homogeneous particles, in other words,"atomism," which science in the twentieth century modifies.The parts themselves, considered without regard to theirposition in the whole event, are nohing. The reality is theorganism, the situation as a whole. The unity of a tree isvery different from that of a machine, and even physicistsare beginning to suspect that they also deal with the formerkind of unity. The effect of this change of view upon educa-tion is difficult to predict.�Everett Dean Martin in "TheMeaning of a Liberal Education."The most important thing in the world is a belief in the

reality of moral and spiritual values. It was because we lostthat belief that the world war came, and if we do not nowfind a way to regain and to strengthen that belief, thenscience is of no avail.�jR, A. Millikan in "The Significanceof ’Radium.’1

Distrust of speculation often serves as a cover for loosethinking.�A. S. Eddington, before the Bmtish Associationfor the Advancement of Science, 1920.There is probably no subject in all science which better

illustrates the importance to the entire world of researchin pure science, than does X-rays. Within three monthsafter Roentgen’s fortuitous discovery. X-rays were beingput to material use in a hospital in Vienna in connectionwith surgical operations. Had Roentgen deliberately setabout to discover some means of assisting surgeons in re-ducing fractures, it is almost certain that he would neverhave been working with the evacuated tubes, inductioncoils, and the like, which led to his famous discovery.�F, K. Richtmyer, American physicist, in "Introduction toModern Physics"

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ELEMENTARY MATHEMATICS IN COLLEGES