11 096 1E17 At rHOR Pec've, W. D., Ed. Mathematics in ... · 1.:1)111()It'4 PREFACE. Tilts Is the sixth or a series of earhook, N10;011111. coloirii of Te:tehers Of Tathematies bevan

DOCUMENT RESUME

11 096 1E17 SE 018 467

At rHOR Pec've, W. D., Ed.T:rL7 Mathematics in Modern Life. National Council of

Teachers of Math2matics, Yearbook 6 [ 1931].FISTITUTION rational Col,ncil of Tfache::s of Mathematics, Inc.,

Washington, D.C.T)113 DATY

205p.A7AILABLF FROM National Council of Teachers of Mathematics, Inc.,

1905 Association Drive, Reston, Virginia 22091

F: RS PRI( 4'

PESCRIPIOR-,

ILFNTIFI"PS

MF-$0.75 HC Not Available from EDRS. PLUS POSTAGEAariculture; 3iology; Humanization;*Interdiscipl..nary Awnoach; Investment;*Mathematical Applications; *Mathematics Education;Physics; RefetencP Books; Religion; Social Sciences;Statistics; *Use Studies; *YearbooksPolygons

AESTRACTChapters present the relation of mathematics tc and

+he usr of mathematics in each of the following areas: socials.cienc biology, religion, investment, agriculture, pharmacy,stati:_ics, and physics. In addition there is a chapter abouthumanistic Learings of mathematics and a chapter explaining theaesthetic values ,af 4ifferent polygonal forms. (LS)

Ti4tz. az=MISSING FROM THE DOCUMENT THAT WAS

SUBMITTED TO ERIC DOCUMENT REPRODUCTION SERVICE:

1.:1)111()It'4 PREFACE

Tilts Is the sixth or a series of earhook, N10;011111

coloirii of Te:tehers Of Tathematies bevan to publish in 1926. Thefirs, &nit %\ it h siii.voy of progrpss ill the past Twooty_tirt,Years," the second vith -Curriculum Problems in Tending.; 111at he-

the third twilit "Sch.( led ','.'firs in the Trachitur of:\lathematies," the fourth with "Shmitivant Changes and '1'1't'i1tbin the Teachint; of lathentaties Throtwhout the World since 1910,"and the fifth with "The '!'ettelting of Cleonictry." liotind copies of:111 but the first of these '..ettrboolis van still be scetircd from thelitir nti of Puhlications, Teachers Collet.te. Columbia University,New Vorli. for $1.75

Yearbook, iinvo heel, well %Trek"l nail have no doubtbeen the source of much 11(.11) to O.:tellers of mullion:dies and toothers intereA(11 in the mathematics held. The sneers. of theseprevious Yearhoolis Iris convinced the N1ltiona1 council of t ht,

desirahility and visdont of rontintting the series. The Sixth Year-book is aceodlindy presented in the hope that it will be helpful notonly to secondary teachers hut to intelinwnt lainc 1 as N1011.

'111(' purpose of the hook is to set forth as completely as possibleill the :mace 1111(itted the place of mathentaties in modern life,Other elutpters, treatint; (If subjvets like 11:011, unities and F.ngi-neerint...., for o:a1,101(., initdit have been inducted, but rie place ofmathematics In such fields is obvious, yo it Nvas decided hest not toinclude them.

I 11:411 to express my pronal appreciation tts well as that of theNational Council to all who have contributed to the Yearhool; orwho in any way have helped to mal:e it what it is.

VI

W. D. ltEnvE.

CONTENTS

ClIAPTISK AO'1, TILE APPLICATION OF MATHEMATICS TO THE SOCIAL.

SCIENCE:4 . bliing Fisher 1

A Purhond Trilmt --(;i1)1pe Contribution!, to Science.--Matliptivities in Economic 'Theory.Snoothing of StatisticalSeries.----Correlation Anitlysi:t of Probability.

II. MATHEMATICS IN BIOLOGY . J. Arthur Harris 18

Introduction.-1. The Fend:mental Characteriet lea of Mathe-matics as Related to the Physical and Biological Series...Premixes and Conclusions The Natural Sciences.The Biol-ogist. II. The Claims of Mathentaties on the Attention ofBiologists. Claint of Pure Mathemativii,Claims of Mathe-matics on the Working Biologist.--The Claim of Sem ice inOther Physical Sciences.The Evolution of Biology and theInfluence of Quantitative Methods.The Two Fronts of theAdvance of Mathematics into Biology. -- III. The More GeneralService of Mathematics in Biology.Certain Limitations andServices of Mathematics in Iiiology.Two Points of Ent-phasis.Sumnitry.

III. THE ITUMANISTIC 'BEARINGS OF AIATIH .

Cassius ,1. Kcyser 36Mathematics and Humanism.--Mathematics Defined Types

of Ilantanism._Tts pti,,,(. Meaning of Ifmnanism.Hutnan-istie Education.Grat Permanent Facts of Life and theWorld.--Nlathematies and the World of Ideas.Attitudes inMathematical Study.-Illimanistie Value of the lk relining At-titudes. Mathematics tine of he Ilumanities.Matinstatiemand the Universal ('oni.erus of Man.

IV. AIATIIEMATCS AND 143.10ION . Eugene Smah 53The Bonds Between Them. 'Corms Consiffered.---The In-

iinite.Tht Changing Bases.The Play of Imagination.Staley and Time.Obscurity of Language.Sonic Effects ofMathematical Study.

V. ME AIATIMAIATICS OF INvixrmENT . William Z. Hart $31

I. Intodution.--Extent of the Discussion.---MathematicalPrerequisites.--Iiistorical Background.---The Unfamiliarity ofthe Subjeet.Outlints of the Simple Interest.

vii

viii cosirEnsPAus

and 1)isrount.-n Alghrr.ir Troatnivnt of Simple. In-trryst.Simpli. 1)isetaint.-111. (*unwound Interest.-1)efinitiottof Compound Inturest.Notitittal and EtTertwe hates .Tht,('ompounti inturest Formula Comment on the Fund-mental

Interet fur 1:1': el lowil perioas.vo.knwn ihirs and Tim,,,..__I.:(toatioo4 AnnuitiesCertain --Annuities... 'resint Value and Atnotatt of an An-11110.Antolit Formulas --Amortixatio)k of it 1)cht........1),,tt.r.Initiation of it Final Amortization Pamnt.Sinkitut Funds.---liontis.--Ilett.rmitettion of Bout c31wiosion.

VI. :\IATIIENIATIS IN . thirry Roe 88Introductiott---Valut. of ?fathentaties in Auriettlture --The

Timing < of Fuming -1111ateil InthiNtr, and ('ommeree.--TheItuol Holm, Problem. --Attrieultural Science and t)rtritnigNI

-ALtrivultur:11 Engineering.--Filiteation in Mathematiesfor Attrieulture.-Ilistorical Ittroset.A Sperialitted Nfolo,maties foe Aqiettlture..-1)iffirtilties Ettrountered.-Selection ofTeaelters....Sulueet Matter Requirements.- -Arithinetio.--Alge.bra,(leottletry.- C-detiltts in High School Courses ---Value ofrttity and tho t 'jut Point. of View.nitheinaties Insrtt-arahlr from Ifttotan xperienee.--I'nity of the Science the11 to soi.(1.,s.- 1'00 lientanded h the Youthful Ntind.

'1).pe Problems Olt.tolete.Nlet.tintr the Spirit of Youth.A Few Well Chosen l'roblents

VII. IkIAIINI.AICS IN PHARMACY AND IN 1)110FER-

Eflward Sprase 102introdurtion.Th Motheninties Curriculum in the Profes-

sional s in NtAthentatics,--Applied Matleretnatieg of Pharittey !oul the Allh41 ProfessionsThe Prep-aration Stud. tus Should Itav.

VIII. *A1.1TIIINI.N.TICS AND STATISTICS . Helen Walker 111I. Nttliod in I're,eitt-1)ay 'Thinking. Importance

of s:ttti.:tiv:11 Ict 11,)(1.---Fm, of Norrial 1)at:i --.H1.10tion ofStat: tied Mrtholl and StatiNtical lietttretttati -Iint NIthod and Mathetteities.--Thp Nature of Statis-tie11 Infrettre.II Nilthenr,tic.il Training for Stti,tiealWork.--A 1)ilt.Inttut ----Ntithentatir.11 Plparation.--11I. Topics:in Arithmetic :Intl .khzlira \t-flott for Beginning 3uttis-tics.Attitutles and Ilabits.Information and Sin-tiAiettl 111(1110(1 for High Selity,1 Sttulnht.--A Cliallmige toTeahers.---Suggested Miortials,The Outlook for Instructionin Statistics,

IX. I\IAITENtATIrs Ix PHYSICS . II. Emmett Brown 136Natural l'hilosophy.The Quantitative Nature of ;Modern

High Sehoid Physif s.Tha Movement Against Mathematics.

CUNTTS ix

PAW'

1C111 of Nillin'111:111es in 111 School Physic/4. -M% N-1411%11'11 PhY:vs ,1 ow is t.f Iiithe-matil:4 in 1. Engy king mid

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1311,..sivg ,\1 itt tilt.., 1, Alvel.a. -2. Other IlighSdlool 11111. lit iiirv. Our. 1.0. .1 Mathriliaties

X. Vol ) ioNA1. 1.4()ItIs (icorge Birk h0,17 10$1. Piviinliwiry 1t, twit, 3.

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Thi. I. of l u a 1.11 17. Summary.

MATHEMATICS IN MODERN LIFE

THE Apru( ArrioN OF MATHEMATICSTO THE SOCIAL SCIENCES *

IRVING FISHERl'ufe i'niper.vity, New Haven, Cann.

A Personal Tribute. It :lily not be amiss to precede what Ihave to say On Mathematies in the Social Sciences by a reminiscentstatement of my personal impressions of J. Willard Gibbs, whosepupil I was foity years ago. J. Willard Gibbs towered, head andshoulders, above any other intellect with which I have come incontact. I had a I, -en realization of his greatness even in thoseformative years in Yale College and the Yale Graduate School,But this keen realization has grown even keener as the years haveswept by, not only because of the increased evidence of the funda-mental value of Gibbs' Nork in his own chosen field but also be-cause in my own consciousness, after so many details have droppedfrom memory, there persists all the more clearly the strong im-pression which Gibbs' personality and teaching made upon me,

In saying this I do not think I can be accused of undue enthu-silm simply from the loyalty of a pupil to his teacher, especiallyru vio of the statements of Lord Kelvin and others, which virtu-,:lly rank Gibbs as the Sir Islay., Newton of America. Lord Kelp insaid w ain visiting at Yale, a few years ago, that "by the year 2000Yale would be best known to the world for having produced J.Willard Gibbs,"

One of the most striking characterizations of Gibbs was ro-cently made by 1)r. John Johnston, now with the United StatesSt( el Corporation. then Professor of Chemistry at Yale, in his ad-Orees on Gibbs .ielivered at Yale University two years ago. Ilestand tlp_t no result of Gibbs' work had yet been overthrown, andthat, in this respect, Gibbs seems to stand unique and supremeamong the great scientists.

Adaptd by irmIshion from iultett of the American Mathiatteni bo'Iety,April 110), Tito sovi.th Josiah Willard (111)11K Lotur, road at Des Moines, Derittbr 31, 11129, hofore it ,14)1111 ssalon of th .%Inerlean NlatboinatIcal Society andtho Amrienit A,hoelat Ion for the Advancement of Science,

1

2 THE SIXT i Y EARBOoK

The English physical chemist, Professor F. l i. Dolman, has paidthe following tribute to him

Gibbs ranks with men like Newton, Lagrange, and Hamilton, who by thesheer fora and power of their minds have produced those generalised state-ments of scientific law which mark epochs in the advance of exact knowl-edge, . The work and inspiration of Gibbs have thus produced not only agreat science but also an equally g-rat. practiee. There is, to-day, no greatchemical or metallurgical industry that. does not depend, for the developmentand cumrol of a great part of its operations, on an understanding and appli-cation of dynamic chemistry and r.he geometrical theory of heterogeneousequilibria.

Professor Ostwald said, in the preface to his t.erinan transla-tion of Gibbs' thermodynamic papers in 1892:

The importance of the thermodynamic papers of Willard Gibbs can bestbe indicated by the fact that in them is contained, explicitly or implicitly,to large port of the discoveries whirl have since been made by various inves-tigators in the domain of chemical and physical equilibrium and which haveleu to so notable to development in this field.. . , The contents of this workare to-day of immediate importance and by no means of merely historicalvalue. For of the almost boundless wealth of results wHch it contains, orto which it points the way, only a small part has up to tho present time 1892been made fruitful.

Sir Joseph !Armor said of the work ,f Gibbs:This monumental memoir On the Equilibrium of &Wow MUM Substances

made a dean sweep of the Kmiec', and workers in the modern experimentalscienco of physical chemistry have returned to it again and again to findtheir empirical principles forecasted in he light of pure theory and to derivefresh inspiration for new departures.

We think no less of Gibbs' greatness because he himself showedso little consciousness of it. He must have realized the funda-mental diameter of his work. But his pupils remarked his pro-found modesty and often commented on it. His chief delight wasin truth-seekhig for its own sake, and he was so intent on thissearch that he had no time even to think of c mphasizing the origi-nality or value of his own ldditions to the great vista of truth overwhich his mind s ..ept. Doubtless he often did not know or greatlycare where the work of others ceased and his own began. Ile didnot always wade through the literature which precried his ownscientific papers. etneinher hearing him say that when he wantedto verify another man's result:, he ,:sually found it easier to workthem out for himself than to follow the other man's own course

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PIVI00: NI S.11,1,VIVIIIIVIV

8 THE SIXTH YEA1U

that there are so few who are trained in both lines for stall study,and this particularly applies to any applications of Professor Gibbs'vector analysis, It vector analysis should become more widelyunderstood and used by students ill the social sciences, doubtlessit would In' Ilion' generally utilized, at least as a vehicle forthought,

Oceasionally, and increasingly, the ideas and notations of thedifferential a1.11 integral calculus are applied by mathematicallq*(11111iStS 11111 Itt 1St jai Wit, of t'Intr:41`, most of the mathe-matics employed 111 the social sciences consists of simple algebra,There is a saying, which, by the way, was quoted by Gibbs in histoldess on NIultiplt, Algebra, that "the human mind has neverinvented a labor-saving machine equal to algebra."

There are several fairly distinct hranehes of social science towhieh mathematics has been, or may lh', ;ginned. The chief ofthese may be distinguished as (1) pure economics, (2) the "smooth-ing- of statistical series, 13) correlation, 4111,1 (4) probabilities, allof which overlap to 50111V Oa .

own chief interest in social science, from a mathematicalp, int of view, has been in the first of these four eyoups, puretheory.

1. Mathematics in Economic Theory. \\lull I began mywork in this field in 1891, mathematies in economie theory waslooked at askance, despite the fact that ninny years before, as earlyas 1838, Cournot had written his brilliant Researches into theMathematical PrIniples th( Theory of Wealth. This hook latergreatly stimulated Professor Edgeworth of Oxford and ProfessorMarshall of Cambridge, and to-day is ranked among the economicclassics. The same may be said of Jevons* Theory of PoliticalEconomy, published in 1871. But in 1891, when my own evonomiestlidies began, even the work of Cournot was almost unknown toeconomists, and that of Jevons was little used. If one \yin turnthe pages of the main economic literature of 1891 and earlier, liewill find practically no formulas and no diagrams. But \\*alrasand Paeto in Switzerland and Pantaleoni and Baroni in Italy,Edgeworth and Marshall in England, Westergaard and Wicksell insc:uplinavia, anti a few other students in other countries were usingand del:tiding the new method.

When, at the request of Professor Edgeworth, I read a slightlymathematical paper on the Mechanics of Bimetallism before the

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6 SH.)XIIJS XI r.)IJ4VIC :'11IJ. I4

10 THE StxTH YEARBOOK

Jevons altnost at once arrested public attention by his brilliant lucidity andinteresting style.. .

A training in mathematics is helpful by giving command over a mar-vt.nouslY lcr$c and exact language for expressing clearly some general rela-tions ft/PI some short processes of economic reasoning; which can indeed beexpressed iu on:inary language, but not with equal sharpness of outline. Anti,what. is of far greater importanee, experience in handling physical problemsby mathematical methods gives a gray. that cannot he obtained equally wellin any other way, of the mutual interaction of econoic changes. The directapplication of mathematical reasoning to the discovery of economic truthshas rt (uny rendered great servies in the hands of toaster mathematiciansto the study Of stati4tical averages and probabilities and in measuring thedegree of eonsilienc between eorri imed ,tansiical tables,

NlathVIllatifS SNITS economic theory in supplying such funda-mental concepts based on t he differential calculus and also throughthe process of differentiation solves problems of maxima andminima, as in the simple process of determining formally what isthe price that the traffic will hear in order to make profits amaximum.

The chief realm of economic theory to which mathematicalanalysis of this formal kind applies is that. of supply and de-mand. the determination of prices, the theoretical effect of taxesor tariffs on prices. The results. cannot always be reduced tofigures but are often useful in terms of mac inequalities.

For instance, among the chief theorems .hown mathematicallyby Cournot are the following:

That a tax on a monopolized article will always raise its price,but sometimes by more and sometimes by le.s than the tax itself.

That a tax on an article under unlimited competition alwaysraises its price but by an amount less than tli tax itself,

That a tax proportional to the net income if a producer willnot affect the price of his product.

That fixed charges among costs of production do not affect pricenot' do taxes on fixed charges.

That opening up free trade in a competitive article between twopreviously independent markets may decrease the total product.

Among the most surprising paradoxes discovered by the mathe-matical method is one shown by Edgeworth, that if a monopolistsells two articles. say first and third class railway tickets, for whichthe demand is correlated, it may be possible to tax the third classtickets. at a fixed amount. each. with the result that. the monopolist

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14 THE SIXTH YEARBOOK

bonds, the formulas underlying the bond tables used in everybroker's office.

While them development of mathematical economies f'om thetheoretical side has been steady and impressive since I was a youngman, it has by no means been so rapid as the development of thethree other branches to which I have referred.

2. Smoothing of Statistical Series. "S'moothing" statisticaldata, the fitting of formnlas and curves to statistics, has. of course,been one of the aims of statisticians for many generations. Inthis way we have derived our mortality tables, the basis used byactuaries for calculating life 'insurance premiums.

I understand that the second J. Willard Gibbs lecture was byRobert. Henderson on Life Ingurance (o a i'4ocial Science and as aHathematical Prob km, The importance of this branch of oursubject does not need to be emphasized in an insurance center likeDes Moines.

Actuarial science is. of course. a development of the formidasfor capitalization or discount., particularly as applied to annuities,combined with the introduction of the probability element. basedon mortality statistics. It is essentially an analysis of interest andrisk. It could he, and perhaps some (lay will be. applied to othereconomic problems besides life iLsuranee, as soon as statistic's areadequate for assessing risk numerically in other realms than hu-man mortality. In fact, one of the crying needs of economic scienceis a reliable basis for evaluating risks.

i'oneurrently with actuarial science has developed a scienceof mathematics of mortality in relation to population, extendingat least back to the days of William Farr, Superintendent of theStatistical Department of the Registrar General's Office of Englandhalf a century ago. To-day this science has been further devel-oped by Knibbs of Aust lia. Lotka and Glover in the UnitedStates, :111(1 others.

Recently, with the development. of statistics of industry. theart of curve fitting, by mathematical methods. has grown veryrapidly, and examples of it will he found in many current issues ofstatistical journals. I :un, thyself, with a collaborator, Max Sasuly.working on a new method of curve fitting aimed to avoid the useof any preconceived formula but letting the statistical data them-selves write their own formula, so to speak.

One important phase of curve fitting whivh links it closely

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LI SHDNI:Miz IVIDOF :,1,LVIViiII,LVV1

.11 ATI 1 EMATICS IN BIOLOGY *By .1. ARTI11711 II.11t1(1:4

nireesit y 0j .11 innesota, Minneapolis, M

':Tkioduction. What is mat hematies? And what right hasmath( to obtrude itself upon the attention of biologists?

Since mathematirs is a very old science, inextricably bound upin its historical development with logic and philosophy on the onehand and with astronomy on the other, it. is perhaps impossible togive a concise definition which would satisfy the workers in allits fields, theoretical and applied. It is easier and more pertinentto our present purpose to indicate some of the characteristics ofmathematics which make it. an essential factor in the more ad-vanced stages of the development of all other science. This maySCIVe as a preliminary to an outline of the claims of mathematicsto the attention of biologists, based on a considerations, onservive in other natural sciences, and on the contribution which ithas already made to the advancement of biology.

I. THE FUNDAMENTAL CHARACTERISTICS OF MATHEMATICS ASRELATED TO THE PHYSICAL AND 13IOLOGICAL 'SCIENCES

Premises and Conclusions. It is one of the characteristics ofmathematics that, starting with certain axioms, postulates, or as-sumptions, it shows the way in which conclusions may he deducedfrom these premises. The mathematician does not necessarily claimabsolute certainty for the physical validity of his conclusions, buthe believes profoundly that it is possible to find groups of axioms.sets of a few propositions eachsuch that the propositions of eachset are eomnoille and that the propositions of each set imply other

The late Professor Barris was invited to write a chapter for this l'arbook onNiathematies ill Biology. lie gladly accepted the invitation and boil started toprepare his material %Olen he died suddenly after an operation for appendicitis.It seems fitting that the should hare something from. Professor Barris' pen. Weare aeeortlim, Cy reprinting by permission this article adapted from Thy setentyleMonthly for ...ognst 1112st. See also Barris. .1. .trthur. '"Ilte Fundamental NIattie-matical Regal. Inent.1 of Biology." Ameriron heintrt iv& MottlIthi. : 1 Tti-1;18.--Tiii: EDITOR.

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MATHEMATICS IN BIOLOGY 21

Claims of Mathematics on the Working Biologist. Let usconsider in outline the. claims which mathematics has On the atten-tion of the working biologist.

The first two may be stated in very general terms. They will,110WeNTr, be developed in the discussion of special claims which isto follow.

The most general contribution of mathematics to the naturalscienees is the affording of an exact and easily worka)le symbolismfor the exiaession of ideas. The progress of science depends verylargely upon the facility with which facts may be recorded andrelationships between them considered. In their bearing upon thisrequirement of the natural sciences it is important to note thatan essential characteristic of mathematical methods is that theyeconomize thought. The notation of the mathematician affords themaximum precision, simplicity, and C011eiSellOSS. The worker innatural science finds in mathematical literature a highly perfectedsymbolism which lie may use without developing one of his own.

But while a convenient notation is the most general contribu-tion of mathematics to the natural sciences, it is neither the onlynor the most important one. In the natural sciences it is essentialthat accurate observations and exact measurements be interpretedby sound processes of reasoning. It seems logical to assume thatthe biologist may profit by the centuries of experience of the mathe-matician in the drawing of inevitable conclusions.

These claims are so general that we may properly turn to thosebased on the specific acwinplishments of mathematics in the physi-cal sciences and in biology itself in substantiation of our argumentfor its wider application in biological research.

The Claim of Service in Other Physical Sciences. Therecord of service of mathematics in the physical sciences is an out-ttinding claim on the attention of biologists.

In the past, mathematics has been an integral part of the scienceswhich we are accustomed to regard as the more highly developed.of all flit.. is physical as distinguished from biological in the growthof our civilization. The most determined critic of the applicationof the mathematical method in biology dares not contemplate theconsequen-es of a :Maxwellian demon snatching from our scientificliterature and from the minds of our chemists, physicists, engineers,and economists the mathematical formlas \Vnichi underlie theroutine of our daily life. In a few weeks long-distance com-

22 THE saTH YEARBOOK

munications would cease, the vehicles of transportation would bemotionless, factories would close, and urban population would facestarvation. As Professor A. Voss said in 1908, our entire presentcivilization, as far as it depends upon the intellectual penetrationand utilization of nature, has its real foundation in the mathe-matical sciences,

If reasoning by analogy is ever justified, experience in thephysical sciences would certainly seem to afford sufficient evidenceof the necessity for the extensive introduction of this powerful toolof research into the biological sciences.

The argument that ,iologists should emulate the workers in thephysical sciences is strengthened by the fact that biological phe-nomena are the most nearly infinitely complex of all naturalphenomena. This is necessarily true because the internal structureand functioning of the organism and the effective environmentalconditions under which it must live and reproduce comprehend amaterial fraction of the physical and chemical complexities of theuniverse. Before the more complicated biological phenomena can begrasped in any but the most circumscribed and superficial way bythe human mind, they must either be analyzed and simplified byexperimental control or expressed in the mentally intelligible termsof mathematical summaries or generalizations.

It may be urged that the method of dealing with large numbersof measurements is not that of the physicist or of the chemist whofrequently works with minute samples under carefully controlledconditions.

The reasons for the differences in methods are two. First, thestudent of molecules has the advantage of working with less com-plex materials and under more readily controlled experimental con-ditions. Second, the physicist or chemist already has his moleculesor ions massed and can investigate them and draw conclusions con-cerning their properties from his examination of the properties ofhis volume of gas or solution. The biologist must begin otherwise.He must collect and determine the characteristics of each indi-vidual of a large sample in order to express the characteristics ofthe whole population in mathematical terms.

When ologists have had the necessary preliminary training,they will realize that, for many of the phenomena with which they.have to deal, the most easily comprehensible and the most usefulmethod of description and analysis is the mathematical. In the


past, biologists as a class have been in reality hostile to the intro-duction into their science of the methods which have proved theirworth elsewhere. I know this to be true from long and bitterexperienve. Instead of being eager to place biology alongside ofphysies anti chemistry in the ranks of the exact sciences, biologistshave seemed not merely to excuse, but actually to take pride in thedistinction which has been drawn between the so-called exact andthe so-called descriptive sciences.

While the historical attitude of the biologist is not excusable, thefault has not been entirely his. With most men mathematics isiike a well-the .leeper they go in the less they see out and about,Mathematics may quite properly be an end in itself, but in biologyit is strictly a means to an end. While mathematicians have inthe past, been eager to serve workers in the physical sciences, andwhile mathematics itself owes a large debt to these sciences,mathematicians have not for the most part felt, it worth while tocome to the assistance of biologists. Mathematicians have oftenasserted the need of mathematic's in the biological sciences, but theclaim has too often been made in all cx cat hedra manner by those\vho, while perhaps qualified to speak of things mathematical, la. :ebeen relatively little fitted to discuss the needs of biology. Whilebiologists have been entirely too slow in recognizing the needs oftheir science for the mathematical tools, they lase shown flu,.practical good sense which characterizes those whose minds havecontact with matter by refusing to flock to the mathematicians'standard until shown by concrete examples that the mathematicalmethod has real applicability in biology. Thus the burden ofproof has largely been thrown upon a few workers of greater vision,\\ all the inevitable result that progress in the application of mathe-matics in biology has been slow.

P.rogress has been slow, but progress there has nevertheless been.The Evolution of Biology and the Influence of Quantita-

tive Methods. The natural sun nces ali had their beginnings inobservation and speculation. Careful description of the observedphenomena, then furnished a basis of interpretation by comparison.I:xpurinientation, which requires not merely controlled conditionsbut measured consequences, followed observation and description.Filadly quantitative measurement, calculatio n. and the formula-tion of mathematical laws have characterized the highest stage ofscientific development.


These stages in the development of the natural sciences are, tobe sure, neither wholly distinct in nature nor sharply separated intime. The methods of the later stages have in some instances beenanticipated by investigators who were in advance of their con-temporaries. It would be unfortunate indeed if men of science didnot at all times avail themselves of whatever is best in the methodsas well as in the results of those who preceded them. Notwith-standing the difficulty of delimiting the various horizons, as ourgeological friends might feel inclined to designate the deposits ofscientific literature of these periods of differing dominant purposes,the sequence is in full accord with historical facts.

The old physicist who defined the biologist as "a man withscientific aspirations and inadequate mathematics" would find, iflie looked over a fair sample of current biological literature, thatnot only has the space devoted to quantitative data increased enor-mously during the past few years, but that there is a steadily grow-ing effort on the part of biologists to express in concise formulas theresults of observation. Unfortunately, biology in most of its phasesstill lacks the quantitative data, anu biologists in general want thetraining in mathematical analysis which is essential in exact science.Nevertheless the tendency of the times is unmistakable; the de-mand for quantitative work is more and more domir. mt in thebiology of to-day.

The most forceful argument for the wider use of mathematicsin biology is furnished by the service which mathematics has al-ready rendered in the biological sciences. Let us consider this morespecifically.

The Two Fronts of the Advance of Mathematics intoBiology. Progress in science depends upon evolution of methodas well as upon the accumulation of the data of observation, ex-perimentation, and measurement. The progress which has beenmade in the development of biology as a quantitative sciencethrough the introduction of mathematical methods is in its presentstage the resultant of various factors, which can be understoodonly when considered in their relation to the evolutionary history ofscience in general and of biology in particular.

This evolution of the natural sciences is admirably illustratedby the history of biology. Observations and speculations beganwith primitive man. If a desire to record what has been seenformed a part of the motives of those who bruised crude figures


on the walls of caves, descriptions began with or before the periodof written language. Some attempts at classification were made ata very early period in man's cultural development, but we areaccustomed to think of the great era of description and classifica-tion as initiated in their modern form by the work of Linnaeus.This period was also one of detailed geographic exploration.Breadth of exploration doubtless tended to stimulate intensity ofinterest in description and classification. The activities of thesedecades resulted in toe storing of great museums with carefullypreserved and minutely described specimens of plants and animals,in the publication of elegant icones which are among the master-pieces of artistic book-making, comprehensive monographs of everylarge genus, encyclopedic summaries of phyla and kingdoms, andfloras and faunas to the end of long vistas of library shelves.

Simultaneously with the latter decades of the period of de-scription and classification of organisms, both living and fossil,began the development of anatomy and embryology. both macro-scopic and microscopic. These latter were indefatigably pursuedby an army of workers whose investigations were so comprehen-sive that the younger and more restless spirits began to fear thatthere would be no worlds left for them to conquer.

With such a wealth of descriptive materials at their disposal,.t was inevitable that serious attempts at interpretation should bemade. Speculation as to the observed phenomena was largely re-placed by effort at interpretation based upon comparison. "It isdescriptive but not comparative," was the criticism of a volume laidbefore the elder Agassiz. The dominance of the comparativemethod over a considerable period of the more recent history ofbiology is attested by the presence of the word comparative inthe titles of a number of institutions and journals.

With taxonomy, comparative anatomy and embryology, his-tology, and cytology well outlined, biologists found themselves freeto extend to other fields the methods which had heretofore beenlimited to physiology. Experimental morphology, experimentalembryology, and experimental evolution are terms which illustratethe degree to which the experimental method has dominated bio-logical investigation (luring the last few years.

a) The Influence of Physics and Chemistry. As soon as biol-ogy, in the course of its evolution, had passed the purely observa-tional and descriptive stage and beTome an experimental science, it

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HOOMIVd.A. ILLXIS 'JILL 96


ful in physics and chetniFtry will become increasingly significantin biology.

b) The Ri Re of Biometry. The penetration of the mathemati-cal leaven into the biological lump through the medium of physicsand chemistry has been so gradual and so little associated with thenames of any individual workers that it has taken place without.biologists as a class being, acutely aware of the profound changein their science. The case is quite different with the second greatline of advance of the mathematical methods into biology. This isdirectly traceable to the development. initiated by Francis Galionand strenuously carried forward by Karl Pearson, of mathematicalformulas suitable for the analysis of the highly variThle data ofbiological observation and measurement; and to the applicationof these methods to a wide range of biological and sociological prob-lems by the biometric school.

While the biometric methods were developed primarily for thestudy of phenomena which are so complex that they cannot begrasped by the unaided human mind or which cannot be readilysubjected to experimental control, they are now being advanta-geously applied to the results of experimentation. Biologists willdoubtless some day realize that experimental results must receivemathematical treatment for ',heir full interpretation.

For the present. there are many who stubbornly refuse to see.We are sometimes told that the biometric constants are merely

a useful moans of expressing results. The idleness of such an as-sertion will he apparent from two simple illustrations.

All mankind has had the opportunity of observing the staturesand other physical characteristics of husbands and wives. Yet itremained for Pearson and his group to slow that there is a high de-gree of assortative mating in man. Why was this not perceived ifthe correlation coefficient only serves to express what we may learnotherwise?

If the suggestion be made that those individuals who observedhuman husbands and wives were for the most part scientificallyuntrained. the reply is evident. Students by the thousands in thebiological laboratories of the world have observed conjugation inParamecium, but it required the biometric investigation by Pearl.wo-king tinder the influence of Pearson. to show that in the unionthe e is a high degree of similarity in the size of the conjugants.Even after the relationship was clearly demonstrated biometrically,


its validity was denied by at least two eminent zoologists. Ifbiometric methods are a useful means of expressing but not ofobtaining results, why did not zoologists long ago note the as-sortative conjugation demonstrated by Pearl, and arrive at theexplanations afforded by the masterly studies in the same field byJennings?

The answer is obvious in both cases. Unaided observation wasincar-tble of dealing with the problems. They required for theirsolution the application of mathematical methods Of analysis toseries of measurements.

These are by no means unique or exceptional cases. Instancesof the failure of biologists to observe important relationships, evenwith the materials or the data before their eyes, could easily bemultiplied. Examples of the misinterpretation of materials or dataequally open to observation could be readily adduced. The mentallimitation implied is not peculiar to biologists. The inability tograsp the more complicated natural phenomena without symbolismis an inherent limitation of the human mind, fully recognized bypsychologists. That a man should be unable to reason about highlycomplicated phenomena without the use of mathematical formulasis no more remarkable than that he should be unable to seechromosomes without the microscope.

Another criticism frequently heard is that the statistical methodscan only locate problemsnever solve them. The real solution.we are told, must in the end be biological, psychological. socio-logical. as the case may be. If this be true, it is the more im-portant that the biologist, psychologist, and sociologist be them-selves capable of using the mathematical methods, or at least ofcooperating intelligently with those who can. But is the criticismreally valid? The same strict.ure is equally applicable to allmethods of research. After a group of phenomena have been de-scribed and analyzed as well as they can he by any means, otherproblems remain to be attacked by new refinements of method orof analysis.

The assertion is often made that the final results must dependupon the original measurements and not upon their mathematicaltreatment. A full discussion of this criticism would lead into sev-eral complexities. but it is sufficient to answer by a very simpleillustration. The possibility of securing accuracy beyond thepower of observation, or at least beyond the degree of refinement


of the measurements adopted, may be easily tested by measuringa series of objects twice, once roughly and once with great accuracy.The statistical constants of these two series of measurements maythen be calculated and compared. Unless there has been a con-sistent bias or personal equation On the part of the observer whichtends to make all his measurements too high or too low, there willbe a remarkably close agreement between the results of the con-stants calculated from the gross and from the refined series ofmeasurements.

Finally, one of the most common criticisms of the hiometriemethods is that they are complex and difficult to use. We havebeen told seriously by biologists that they expect to adopt thebiometric methods when they shall have been more simplified andhence made more suitable ror practical use. But research does nottend to become simpler with the advance of science. Since biologi-cal phenomena are innately complex. there is no likelihood that themathematical formulas required for their investigation will be sim-plified except in matters of practical technique. Criticism of thebiometric methods on the ground of their difficulty is merely theglorification of the mental lassitude of the critic.

Let us turn from the answering of criticisms to things moreconstructive.

If science is to advance at the rate which we desire, anotherhighly practical consideration cannot be neglected. Many biologi-cal phen mena cannot be subjected to experimental control. Thuswhile the proper study of mankind may he man, human individualsand their relatives cannot be investigated in the same manner aswhite rats and Drosophila. While man may be the most con-spicuous illustration of an organism which cannot he studied in abroad way under controlled conditions, the example is not unique.In innumerable cases the statistical study of masses of data maynot only properly, but must necessarily, replace controlled experi-mentation, I hope to show later that in such cases 'the experi-mental and the statistical method are in essence identical.

Even where refined experimentation is possible the biometricmethods are particularly suited to reconnaissance work. In thesearch for the relationship between different variables the statisti-cal analysis of large masses of comparatively rough data may indi-cate the place in which carefully controlled experiments may andshould be made, Finally, after biological problems have been sub-


jected to as close experimental control as possible, the results aregenerally so irregular as to make biometric analysis desirable.

Let us consider briefly, in review, the claims of the biometricmethods to the attention of biologists.

First: The biometric notation makes possible the expression ofthe results of extensive experience in concise and mentally compre-hensible terms.

This matter of the form of expression is one of far greater im-portance than might at first be realized. Rapidity of progress inany branch of science must depend very largely upon the facilitywith which the data and conclusions of a new investigation can becompared with those already on the library shelves. It is by thereoccurrence of like results that general theories are established. It.is by the noting of inconsistencies and the circumstances underwhich they occur that indications of as yet unsuspected relation-ships are often seen.

There can be little doubt that the rapid advance of physicsand chemistry has been due in no small degree to quantitative andstandardized modes of expression.

If the physicist or chemist wants a solubility, melting point. orconductivity of any substance, he has merely to turn to volumesof constants to find whether it has been determined, and if con-stants are available. whether the recorded results accord with hisown. An investigator has been able to draw upon a common fundof knowledge to a greater extent and with greater ease than inbiology. Thus synthetic work has been facilitated.

In its bearing on the problem of the simplification of scientificliterature, consider for a moment the state in which biology wouldhe to-day had it not been for the Linnaean notation, by whichspecies may be designated by a simple binomial instead of by acumbersome description whenever it is mentioned. The value ofthi:z relatively suceinct notation becomes especially apparent whenwe contemplate the vast harm which has been done to scientificresearch through the unwillingness or inability of taxonomists tom-,intain uniformity of nomenclature. Then in view of what hasbeen accomplished by this relatively simple expedient. imagine therapidity of advance which will he possible when a quantitative modeof expression permits the results of many fields of biological re-search to he summarized in annual volumes of standard constants.

personally. am inclined to look upon the publication of Donald-


son's volume on the rat, in which the experience of a whole insti-tution of workers is summarized in quantitative terms, as a realmilestone in the progress of biology.

Second: The biometric formulas provide a system of probableerrors which safeguards the worker in the formulation of his con-clusions. Biometririans have referred so freely to probable errorsthat critics have facetiously sm,gested that biometry is chieflyerror. But frankly and candidly, if a given set of observations isinsufficient to demonstrate a relationship, is it not better that theinvestigator discover the fact himself than that he should publisherroneous conclusions wi,icli must be corrected by subsequent. re-search'?

Third: The biometric methods not. merely furnish a system ofmentally comprehensible constants and concise equations, suitablefor the description of complex phenomena. and a series of probableerrors which safeguards the worker in drawing conclusions con-erning these phenomena, hit they make possible the investigationof relationships so intricate and so delicate that they are quite be-yond the scope of unaided observation. Here the biometric methodshave a potentiality for service analogous to that of the equipmentof the modern observatory, which is capable of dealing with stellarphenomena that were beyond imagination a century ago, or to thatof modern microscopic equipment. and technique which have givenrise to whole sciences of microcosms which were beyond the kenof Linnaeus. To argue that it is unnecessary to push on into theinvestigation of these more recondite relationships is as contraryto the spirit of science. as reactionary. as to argue that it werebetter to have stopped with Galileo instead of advancing to therefinements of modern astronomy through the development of in-struments and mathematical theory.

Fourth: For many classes of problems the biometric formulasapplied to large masses of data furnish the closest possible approxi-niation to the experimental method of investigation.

The experimental method, as ideally applied. consists essentiallyin the simplification of conditions by rendering constant all butone. This one factor is then varied and its influenee upon theorganism is noted. In certain phases of statistical analysis anessentially identical method is followed, when we determine whatis called the partial correlation between two variables for constantvalues of one. two, or more other variables.


For example, the basal metabolism of tall men is on the aver-age greater than that of those of less stature. Heavy men alsoshow a higher daily gaseous exchange than light ones. 'Rut tallermen are on the average heavier men, and it seems quite possiblethat the larger basal food requirement of taller men is merelythe resultant of the relationship between stature and body weighton the one hand and between weight and metabolism on the other.The bioinetrieian solves such a problem statistically by determin-ing the partial correlation between stature and metabolism forconstant Nveight, i.e., with the influence of body weight eliminated.The experimentalist would have to attack the problem in exactlythe same manner.

Illustrations might be given by the score of the analytical treat-ment of statistical data which gives results of essentially the samenature as those w bleb are attained by the experimental method,often in cases in which strictly experimental technique cannot bereadily applied.

Fifth: The biometric formulas furnish the best means as yetavailable for predicting the value of one variable from another,or from a series of others. This is due to the fact that it is pos-sible to pass at once from measures of interdependence in termsof the un.v. Nally comparable scale of correlation to regressionequations showing the rate of change in terms of the actual workingscale of any variable associated with another, or others, whosevalues are known.

The great theoretical importance of this feature of the biometricmethods will bo clearly realized when we remember that. the testfor the validity of a theory is its capacity for predicting the un-known.

The foregoing treatment in outline may have 1-,en disappoint-ing to those who have expected argument by illustration of specificaccomplishment. The method has berm followed because the bio-logical contributions which have already been made through theuse of the biometric methods are now so large that no one man,even with unlimited space, can be expected to summarize them.This is true, notwithstanding the fact that the number of workerswho have persistently stood by the bioinetric guns during the longand discouraging. years of general indifference on the part of biolo-gists can he counted on the lingers without using all the digits ofthe hands.

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physicist, the chemist, and the astronomer. As yet little progressin this direction has been made in biology, but I ani glad to go onrecord as predicting that before many years have passed experi-mentation will be to a considerable extent guided by preliminarycalculation.

My second point has to deal with a very different matter.Elegance or form has always made a powerful appeal t, 'he

mathematician. As the biologist is forced by the inevitable prog-ress of his science to occupy himself more cad more with mathe-matical literature, its logic, terseness, and elegance of expressionmust have an ilitlUence upon his own standards of presentation.

SUMMARY

Summarizing in a few sentences we may note that mathematicsis driving into biology on two wide fronts.

On the one, physics and chemistry are by virtue of their influ-ence upon biological research forcing biologists to take over themathematics which is an indispensable part of these sciences. Onthe other, biometry is grappling with problems which are not readilyamenable to experimental treatment.

The possible contributions of mathematics to biological scienceare too varied to be succinctly summarized. We must, however,record our entire disagreement with the dictum that mathematicsis only a mill from which no more conics out than was originallyput in. What Are put in are raw data, the significance of which isobscured by all the perplexing irregularities due to morphologicaland physiological variation, to errors of random sampling, and toerrors of measurement. What collies out is a series of mathematicalconstants and equations, epitomizing in mentally intelligible formthe whole discordant, mass of irregularities and smoothing them ina manner to bring out the underlying laws. To assert that thevalue of a bionietrie research is determined by the raw biologicaldata is not altogether unli':e incastiring the value of a Titian bythe grams of pain ,(tii.ired to cover the canvas.

It has been the rali good fortune of Quetelet, Galton, and Pear-son to initiate one of i he great lines of advance in biology. Thesemen will one day receive from biologists recognition as free andgenerous as their great service merits. As for the rest of that littlehandful of workers who have made up the biornetric school, it hasbeen the satisfaction of a few never to have stepped back from the


guns during the long and discouraging years of biological indiffer-ence and opposition.

The ultimate recognition mathematical biology is merely apart of that inevitable and irreversible evolutionary process bywhich biology is to take its place in the ranks of the exact sciences.

THE HUMANISTIC BEARINGS OFMATHEMATICS

By CASSIUS J. KEYSERColumbia University, New York City

Mathematics and Humanism. In order to reduce the hazardof being misunderstood I will begin by giving some indication ofthe senses in which the terms Mathematics and Humanism are tobe understood in the following pages.

Mathematics Defined.' After many centuries of endeavor ithas become possible in recent years to define Mathematics with ahigh degree of precision and clarity. Mathematics may be viewedas a body of achievements or as an intellectual enterprise. I preferto view it as an enterprise, and I define the great term in the fol-lowing words: Mathematics is the enterprise which has for its aimto establish Hypothetical propositions. By a hypothetical propo-sition I mean one that either is stated, or admits of being stated, inthe form, p implies q, where p denotes one or more propositions(called axioms, postulates, assumptions, or primitive propositions),where q denotes a proposition (commonly called a theorem), andwhere the verb implies is intended to assert that q is logically de-ducible from p.

It is common and often convenient to state a hypotheticalproposition in the form: If p, then q. But here one must be onone's guard, for it is obvious that many propositions, though statedin this form, are not hypothetical. For example, the propositionif it lightcns, then it will thunderdoes not mean to assert that theproposition, it will thunder, can be logicaPy deduced from theproposition, it lightens. The test as to whether a proposition ofthe formif p, then qis or is not hypothetical is whether theassertor is intending, or not intending, to assert that q is logicallydeducible from p.

It is to be carefully noted that a hypothetical proposition is

1 For a full exposition of this and kindred conceptions the reader may be re-ferred to my book, The Paotura of Wonder, Columbia University Press.

36

HUMANISTIC BEARINGS OF MATHEMATICS 37

true or false according as the asserted deducibility is or is notpossible.

A mathematical proposition is a hypothetical proposition thathas been established. And an "established" proposition is one thatis so spoken of, so regarded, so treated, by all or nearly all expertsin the field or subject to which the proposition belongs.

The foregoing conception of mathematics gains in clarity byviewing it side by side with the following conception of Science.This term ought, I am convinced, to be defined as follows: Scienceis the enterprise having for its aim to establish Categorical propo-sitions. A categorical proposition is one stating that such-and-suchis the case, regarding no matter what part or aspect of the actualworld. A categorical proposition, no matter what its form, neverasserts logical deducibility, or implication, but, as I have said, al,ypotlletical proposition always does.

By mathematical method I mean all available means estab-lishing hypothetical propositions. Of these means, one isalways absolutely indispensable. I mean Deduction; all othermeans are but auxiliary thereto, mere servants or helpers, neveradequate in themselves. The law is: No deduction, no mathe-matics.

By scientific method I mean all available means for establish-ing categorical propositions. Of these means, one is sovereign,always absolutely indispensable. I mean Observation, all othermeans are but auxiliary thereto, mere servants ur helpers, neveradequate in themselves. The law is: No observation, no science.

In mathematics Deduction is supreme, observation and all othermeans subordinate.

In science Observation is supreme, deduction and all othermeans subordinate.

Types of Humanism. As for Humanism, it is especially im-portant to indicate the sense in which the term is to he employedin this essay, fur that fine old word is to-day used in such a varietyof incompatible senses that, unlike the term mathematics, it cannotnow be said to have a standard signification. And so there areI-nnanisms and humanisms. It will, I think, be a helpful prelim-Mary to signalize some of them.

There is, fur example, the recently much-discussed Humanismwhich Professor Irving _Babbitt ks been endeavoring for two orthree decades to formulate and foster by means of leeturei. essays,


and books, and which he and fourteen other representatives of hiscult set forth a few months ago in the form of a symposium en-titled Humanism and America, edited by Mr. Norman Foerster.In his editorial preface Mr. Foerster tells us that "Professor Babbitthas done inure than any one else to formulate the -roncept ofhumanism," 1111(1 that he is, in Mr. Foerster's opinion, "at the centerof the humanist movement." It is, then, not strange that Pro-.fessor Babbitts Nile in the symposium is that of official definer.Mr. Babbitt teli., us at the beginning of his "Essay at Definition"that definition is "indispensable." Naturally the reader is glad-dened by the prospect of finding here an authoritative formulationof the proper meaning of the great term Humanism.

What, then. is Professor Babbitt's definition of that term? Hestates it in these words: "Humanists are those who, in any age,aim at proportionateness through cultivation of the law ofmeasure." In point of form the defirition good. but what of itssubstance? No mere hitching togethel of such hazy verbal abstrac-tions eau convey any definite idea. Obviously the definition issadly in need of interpretation. The entire essay may be viewed,and was doubtless intended to be viewed, as an attempt at suchan interpretation. And what is the interpretation? It consistsmainly of fragmentary descriptionof scattered bits of descriptionof what Professor Babbitt means by Humanism. Mathemati-cians need not be told that there is a radical difference betweendescription and definition. Of Mr. Babbitt's scattered bits of de-scription some are positive but most of them are negative. Themost revealing of the positive bits arc these: The humanist "maywork in harmony with traditio:lal religion, yet lie says with Pope,

Presume not God ti scan:The proper study Of mankind is man;

his central maxim is 'Nothing too mull"; like .Milton he regardsdecorum as the "grand masterpiece to observe"; his final appeal isto intuition; the basis of the pattern be imitates is not divine butis "the something in nians nature that sets him apart simply asman from the other animals"; humanism manifests itself primarily,"not in the enlargement of comprehension and sympathy," but in"selection," in the imposition of "a scale of values-; like MatthewArnold, the humanist "hates all overpreponderance of singleelements"; he aims at approximat'ng ever nearer and nearer to


"perfect poise"; for the humanist th, universe is sundered by avariety of old dualisms, the psychical and the physical, the sub-jertive and the objective, the properly human and the subhumanor natural, the properly human and the superhuman or super-natural, "law for man" and "law for thing,'' the free and thedetermined; essential for him is the "higher will," here called the"higher inimetliacy," whose function it is to control the "lowerinnnediacy"---"the merely temperamental man with his impressionsand emotions and expansive desires.-

Such are the positive bits of description, here assembled (notm it bout labor) from various sections of Mr. Babbitt's essay. Mostof them have lung been familiar suggestions of the ideal and manyof them are admirable. No doubt they help us somewhat to under-stand what it is that Professor Babbitt's definition of If innani$mis intended to define. We must not be too sure, however, what these

Isitive bits are designed to describe until Mr, Babbitt has in-terpreted them fur us. In interpreting them he has supplementedthem with many descriptive bits of the negative kind designed totell us what Humanism is not. In the light of these negations weperceive that Mr. Babbitt's brand of Humanism is almost incrediblystrict and exclusive,

Amazing, vast, auk., very impressive is the array of human in-trests, points of view, cults, activities, enterprises, personalities,enthusiasms, aspirations, dreams, that Mr. Babbitt, either explicitlyor by implication and with the air of pontifical authority, excludesoutright from the category of things humanistic. All monists (whodeny or question the tenability of the old fainiiiar till:distils. andattempt to View the nuiverse as a genuine cosmos sonnshow involv-ing the unity of Nature and Man), all naturists who believe that"out of the earth the poem grows like the lily or the rose"), allhumanitarians (artuatcd anti sustained by fait h in t he endlessperfectibility of mankind), all romantici.ts, all determinists, allrealists, all the philosophers who regard the One as a "concept"instead of a "living intuition,- all the colleges and universities and()thew educational institutions that "proclaim the gospel of service,"all pragmatists, all psychologists, all devotees of science, all

is texvept specialists ill Mr. Babbitt's variety of Human-; all of these anti yet other kiwis of the unworthy are rigor-

ously excluded by Mr. Babbitt froni his hunianiA iv tTo admit men and women having any essential similitude to such


as Epicurus or Lucretius or Nietzsche or Shelley or Shakespeareor Rabelais or the Walter Paters or the Rousseaus or the WaltWhitmans or the William thuneses or the Benedetto Croces or theThomas Ilardys would he to profane the sacred temple, and sothey are debarred. Debarred also, by the clearest indicia of Pro-fessor Babbitt's conception of Humanism, are such creators ofpsychic light as Spinoza, John Locke, David Hume, Euclid,Newton, Einstein, Willard Gibbs, Lobachevski, Riemann, Gauss,Laplace, Lagrange, Charles Darwin, Herbert Spencer, HenriPoinear6, Louis Pasteur, to name only a few of the greatest of thegreat Unfit.

It is evident that Humanism, as conceived by Professor Babbitt,is too lacking in catholicity, in spiritual amplitude, in magnanimity,to attract any one except. Mr. Babbitt, his disciples, and fellowsymposiasts, who regard it as the sole remedy for healing the cul-tural maladies of the world, especially in America, and as the solemeans for qualifying human individuals to represent worthily, intheir life and work, the great potential dignity of Man. Some havecalled it "academic" or "striet'' or "doctrinal" Humanism. Itmight be described, not inaptly, as supercilious, sectarian, phari-saical, arrogant. "It is," as Doctor H. S. Canby has said, "a veryporcupine hunched up against our familiar world." Speaking of itscentral standard of literary excellence, Mr. Henry Hazlitt haswritten: "We are above all to judge a writer, not by his originalityor force, not by his talent or genius, but by his decorum! Thatis, we are to praise him for a virtue within the reach of any learnedblockhead."

Very different from the foregoing is the type of Humanismdelineated and advocated by Charles Francis Potter in his beauti-fully written, sympathy-winning book Humanism: A New Religionvery different in content, in manner, and in spirit. For Mr.Potter and his kind, "Humanism is faith in the .supreme valueand self-perfectibility of human personality." It might be calledhuman Humanism or nontheistic Humanism because, though it be-lieves in man, it does not believe in a supernatural God. Mr.Potter's conception of Humanism as a religion is probably due tothe fact that he was bred in, and for many years practiced,theology. His religion, however, has recently undergone a greatchange. for he now agrees with Ames that "religion is the con-sciousness of the highest social values" and with Haydon that


"religion is the shared quest of the good life." Mr. Potter's re-ligious Humanism is scientific in the sense that it looks mainly toscience for help in more and more realizing "the twin visions"the vision "of an ideal developed human personality" ari thevision "of an ideal commonwealth made up of such personalities."But Humanism can no more be defined exclusively in terms ofReligion than in terms of Art or Politics or Education or LiteraryCriticism or any other one among the great interests of man as man.

A third variety of Humanism is that portrayed in The NewHumanism by Leon Samson. The hook is hold, richly suggestive,frequently keen, notably omniscient, strangely vislonary, and oftenflamboyant. Mr. Samson's so-called Humanism might be fairlydescribed as loquacious Humanism, for, says he, "There is nothingthat so unmistakably marks a man human as his capacity to talk";or it might be called proletarian Humanism, for "the prized jewelsof contemporary society will be turned to ashes when the pro-letariat lights the fire of life and love on the funeral candles ofcivilized culture"; or it might well be designated utopian or elysianHumanism, for it envisages a planet-wide community of humanswho, having outgrown both war and work and the making andreading of books and all religions and morals and governments andfill other historic or existing institutions, will thereafter devotetheir unbroken leisure ecstatically to endless conversationtohonest, original, infinitely varied and, of course, unfatiguing musicaldiscourse by word of mouth.

The Proper Meaning of Humanism. It is hardly necessaryto say tl.at in dealing with the humanistic hearings of mathematicsI shall have in mind a conception of Humanism vastly differentfrom any of the foregoing varieties. It cannot be defined in termsof Mr. Babbitt's "decorum," "proportionateness." and "law ofmeasure," nor in terms of Mr. Potter's excellent. "religion," stillless in terms of Mr. Samson's wholly fatuous proletarian dream.Indeed I shall not attempt to define it at all. For. as in the caseof ninny another great idea--that of justice, for example, or wisdomor poetry or knowledge or truth or religion or art or loveitssignificance is too immense, embracing too much of life, to admitof being confined in a precise formula. But, though it cannot beneatly defined, it can he described well enough for the purposesof identification and recognition. In respect of brevity, clearnessand comprehensiveness, combined, the hest description I have en-


countered is in the following words of Mr. Walter Lippmann:Humanism "signifies the intention of men to concern themselveswith the discovery of a good life on this planet by the use ofhuman faculties." The Humanism indicated by that description isin spirit, in aim, and in implicit scope, identical with the Humanismwhich, beginning in the fourteenth century, sprang into full life andgreatly flourished throughout the fifteenth century as an essentialpart of f he Renaissance, first in Italy and later in other countriesOf Europe. The term humanist, which then came into use, wasapplied to those men who, by their activity, proelainied the fullrecovery of a very precious and very powerful human sense, onethat had been lost and almost. extliiguished in the preceding longcenturies of submission to external authorityI mean the sensethat humans are, as such, endowed with the dignity of autonomousbeings. potentially qualified by native inheritance 'so judge indi-vidually and independently in all the great matters of human con-cern and, by the exercise of their own faculties, to fashion theirlives worthily.

That sense of personal autonomy is essential to the properdignity of man and it is, as I have intimated, in the central coreof Humanism. The fact is continuously manifest. and frequentlybecomes atticulate, in the activity of the great humanists of theRenaissance. Let me cite one or two examples. One of the mostillustrious humanists of the fifteenth century was Pieo dellaMirandola. in his famous Oration on the Dignity of Man he rep-resents God as addressing Man in the following remarkable words:"The nature allotted to all other creatures restrains them withinthe laws I have appointed for them. Thou, restrained by no nar-row bounds. shah determine thy nature thyself accordhn, to thineown free will, in whose power I have placed thee. I have setthee midmost the world in order that thou mightes the more con-veniently survey whatsoever is in the world. . . . Thou shalt havepoker to decline unto the lower or brute creatures. Thou shalthave power to rise unto the higher, or divine, according to thesentence of th\ intellect." Note Pico's vigorous assertion of theautonomous nature of Man, and observe, too. now perfectly hisutterance times with the following words of another eminenthumanist of the time, Leon Br.ttista Allierti "Men can do allthings if they will." Having in their hearts so living n sense ofpersonal sovereignty it is to wonder that the great humanists of

HUMANISTIC BEARINGS Or MATHEMATICS 43

that period cast off the shackles of ecclesiastical authority, andit is no wonder that they were so eager in seeking, mastering, andemulating all that remained of the literature, philosophy, science,and art that had been Treated by the great humanists of antiquity.

Humanistic Education. I hope that I have now sufficientlyintimated what it is that the term Humanism is to stand for inthe following discussion. As most of those who will read this essayare processibnal teachers I will try to view my subject from thetandpoint of an educator and will deal with it, as an educationalt heme.

By humanistic education I mean education having for its aimto qualify human individuals to represent worthily, in their lifeand work. the great potential digLity of Man. In other words, Imean education characterizeu by the aim of qualifying humanindividuals to discover" orwhat is tantamountto create "agood life on this planet by the use of human faculties." I say"on this planet" because in all times the great humanists havebeen sane enough to concern themselves primarily, if not exclu-sively. with mundane affairs, with means to excellence of life hereupon the earth. The discerning reader will readily see that thetwo statements of aim are virtually equivalent.

It is obvious th..t in humanistic education we are concernedwith the liiirliest and most composite of genuine ideals. "Com-posite" because it embraces many other ideals which, though theyare also genuine. are suborriinate and auxiliary, for genuine idealsconstitute a hierarchy of dignities. In this connection I cannotrefrain from saying, what I have repeatedly said elsewhere andshall never miss an opportunity to say, that genuine ideals are notgoals to be reached but are perfections to he endlessly pursued.Genuine ideals are like those mathematical limits whose variablesapimmeh them ever more and more nearly but never attain them.I know not how to condemn with sufficient severity that all toofamiliar philosophy, for it is now in much vogue, which counselsus to eschew genuine ideals on the alleged ground that, becausethey are unattainali!e, they tend to dishearten and devitalize. Tohearken to that counsel is to turn away from the most powerfulhire!: to excellence. For it is pursuit of unattainable ideals thathas led to the great triumphs of the inenan spirit in every depart-ment of life. It is indeed the proper vocation of man.

In the theory of humanistic education it is necessary to dis-


tinguish between a human being and the mere follower of a humanpursuit. As animate creatures inhabiting a world where we humansare obliged, like the animals, to win our lives from day to day, weall of us are, or are destined to he, in some sense, hewers of woodand drawers of water. And so we all of us require vocational orprofessional training. We cannot escape the necessity of beingspecialists of one kind or another. The ideal of such training, theideal of specialism, is efficiency. But efficiency is not. the ideal ofhumanistic education. For humanistic education aims at the de-velopment and the disciplining of the whole man. And the manbuilding a bridge is immeasurably greater than the engineer; theman teaching the calculus is infinitely greater than the mathema-tician; the man cultivating fields is vastly greater than the farmer;the man painting a picture is incomparably greater than the nrtist.Humanistic edneation does not exclude the ideal of efficiency.What it disowns is the ideal of mere efficiency. The ideal ofhumanistic education is intelligence, emancipation, magnanimity:intelligence regarding the human and the non-human worlds; eman-cipation from every manner of trivial or sordid things, emancipa-tion from provincialism, from fanaticism, from bigotry, fromprejudice, from the multiform tyranny of fear; and magnanimity,largeness of mind and spirit, imagination enough and sympathyenough and reason enough and emotion enough and will enoughto gain and maintain the lordly poise of a freeman amidst all timetrials and frustrations encifintered in a vast, complicate. perplexingworld.

Great Permanent Facts of Life and the World. It is obvi-ous that humanistic education aims to orient and discipline ourhuman faculties, not with special reference to the technical require-ments of any given pursuit, no matter what. but with reference toall the great permanent massive facts of life and the world. Alittle reflection suffices to show that there are such facts.

One of them is the fact that every human beim, has behindhim an infinite past out of which he has come and which containsfor his guidance and edification the records or the ruins of alt theexperiments that man has made in the art of living in the world.It follows that humanistic education will not neglect the disciplinesof the history and the literature of antiquity.

Another of the great abiding massive facts of life and the worldis the fact that we humans are constrained by forces beyond our


control to live, not in isolation, but in human society; that we areliterally born members of a thousand teams with wkich we mustlearn to coiiperate in some measure or perish. It is, therefore,evident that humzuti..tie education is bound to provide disci-pline in political science, in social science, in ethics, and injurisprudence.

A most impressive member of the group of great permanentmassive facts of life and the world is the ubiquitous presence ofBeauty. Beauty is the most precious and the most vitalizingelement in the universe. More than aught else it is beauty thatnot e.21y makes life worth living but makes it possible; for if bysome spiritual cataclysm all the beauty of nature and all thebeauty of art and all the beauty of thought were suddenly blottedout, our human race would quickly perish by depression of spiritowing to the omnipresence of ug,liross. Consequently humanisticeducation will fashion itself in large measure by the considerationthat those who are to be qualified for the discovery or creation ofa good life must needs have taste in. the arts of men and anawakened sensibility to the marvelous natural beauties of Ian i andsea and sky.

Mathematics and the World of Ideas. Among the momen-tous facts with reference to which it is the function of humanisticeducation to orient and ("scipline our faculties I have now, finally,to signalize the stupendous fact denoted in German by the termGrdankentceltthe world of Ideas. For equipping one to deal withideas as suchto deal with them, that is, in accord with the lawsof flint ght, in accord with he standards of rigorously sound think-ingthere is but one available diseipline, and it is that of logicand mathematics. In deference to conservative usage I have saidlogic, and mathematics, though the maturest critics regard the twoas one. It is evident that to he set in right relation to the worldof ideas, though it is not alone sufficient, is certainly necessary, toqualify one to represent worthily the proper dignity of man. Weare thus bound to say that, for humanistic education. logic andmathematics constitute not merely a useful discipline but one thatis indispensable.

Attitudes in Mathematical Study. The fruits of that dis-cipline naturally vary with the attitude of the student in pursuingit. and the student's attitude may be any one of three. He may,that is, pursue mathematics for its own sake or for the sake of


its uses and applications or for the sake of what I shall call itsbearings.

One who pursues mathematics for the sake of mathematics issustained by its charm. Having gained some knowledge of it, liecraves yet more, and what he has gained at any stage equips himfor further gain. If he he a research mathematician, the investig-tions he makes lead him to further investigations. and so on end-lessly. The attitude of such a student. reminds one of the fannerwho. when asked why he raised so much corn. replied, "In order tofeed hogs," and when asked why he wished to feed so many hogs,replied, "In order to buy more land," and when asked why he de-sired more land. replied, "In order to raise more corn." I am notcondemning the attitude, far from it, but, merely indicating it.This attitude of the mathematician is indeed well justified by twoconsiderations. One of them is that. he has great joy in the game.and any one who has felt the joy knows how sustaining it is. Thesecond consideration is that, if mathematicians did not pursuethe subject without regard to its applications or uses. then. whenmathematic-11 doctrines are needed as instruments in the develop-ment of other scientific subjects, the required doctrines would nothe in existence. Moreover, one of the important lessons of historyis that doctrines created for the mere joy of creating them, created.that is, without regard to any question of applicability, are sooneror later fove,(1 to be applicable and thus acquire a secondary typeOf justification. A familiar example of this very significant factis afforded by the theory of conic sections, which was essentiallyworked out for the pure intellectual joy of it long before it foundapplication in astronomy and navigation. A more recent and evenmore striking example is that of the frightfully complicate Theoryof Tensors. established by Riemann and Christoffel long beforethe "idle theory" became, at the hands of Einstein and his fello.:.the "backbone" of the General Theory of Relativity. In this con-nection I must repeat one of the delightful stories told of J. J.Sylvester when he was professor of mathematics in the JohnsHopkins University. One day as he was crossing the campus heencountered a notably practical-minded colleague, who said to him:"Professor Sylvester. what subject are you lecturing on this term?"The great mathematician replied: "On the Theory of SubstitutionGroup:? " "What," asked the practical man, "is the uRe of thattheory?" "I thank God," said Sylvester, "that so far as I know it

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the number-concept, from the rudest beginnings in primitive mind,long before men had learned even the first steps in the processof counting. to the great number-creations of the modern world?The concepts of Integers and Fractions, of Cardinals and Ordinals,of Positives and Negatives, of Rationals and Irrationals, of Rea lsand Imaginaries, of Algebraics and Transeendentals, of Finites andInfinites, these great concepts viewed with the occasions of theirrise, with their struggles for existence, their ultimate triumphs overstubborn opposition, their persistent. hardy growth through thecenturies, their countless diversifications and subtle refinements, theinfinite network of their interrelations and their manifold, alwaysincreasing. practical and theoretical uses and applications, afforda series of scenes that, for any one who has once contemplatedthem, constitute a truly unforgettable and inspiring panorama ofthe march of mind.

Mathematics One of the Humanities. It remains to considerthe attitude of one who pursues the study of mathematics. not forthe sake of mathematics, nor for the sake of its uses and applica-tions, but for the sake of its hearings. By the bearings of mathe-matics I mean the relations of mathematical ideas, processes, anddoctrines to such great. human concerns as are not, strictly speak-ing, mathematical. One who pursues the study with a view to itshearings can hardly fail to discover that, by virtue of its humanisticsignificance and worth, mathematics is entitled to high rank amongthe great Humanities. For what are the subjects that are bestentitled to be listed among the humanities? It can hardly bedoubted that the answer ought to he this: Those subjects have thebest datum to be called humanities which best serve to reveal +henature of our common humanity and best serve for the guidanceof our human life.

It can he readily shown, I think, that, according to the doublecriterion just stated, the claim of mathematics to he regarded asone of the humanities is unsurpassed, Let us examine the mattera little. We must begin by asking: What is the chief mark of manas man? What. is the defining quality or character of the essentialnature of our common humanity? What is it. that serves best to(11:AC1'1111111OP lwings from all other kinds of living creatures?The answer. I think, is this: The chief characteristic mark of manas man is what Count Alfred Korzybski in his hook, The Manhoodof Humanity, has called the time-binding enpacity of human


beings. The fine term denotes that highly composite faculty invirtue of which each generation of mankind is enabled to employthe accumulated achievements of the preceding generations ascapital for the production of yet greater achievements, so that, asthe generations succeed each other, science begets better science,philosophy better philosophy, art better art, jurisprudence betterjurisprudence, ethics better ethics, religion better religion, inven-tion better invention, and so on. By that composite faculty manis set apart from the animals and the plants. It is the secret. ofthe progressability of our human kind. It is the civilizing energyof the world.

Where does the time-binding power of man make itself mani-fest? Obviously it manifests itself in the development of all great

subjects and human enterprises. I now ask: Where is this defining

mark of man revealed most clearly? It is most clearly revealedin mathematics, for in the continuity of the progressive develop-

ment 01 mathematics, running from remote antiquity down through

the centuries and flourishing to-day as never before, the time-

binding power of the human intellect. is not only revealed butrevealed in its nakediasss. I contend that, by this superior dis-

closure of the charaetestic nature of our common humanity,mathematics conclusively vindicates its claim to distinguished

membership in the assembly of the humanities.If we turn now for a momeut to contemplate mathematics re-

garded as a guardian and guide of our human life, we shall find

that the foregoing conclusion is abundantly confirmed. As every

one knows, human activity presents certain great distinctive types.

One of the greatest of these types is that which we call logical--thinking, not merely thinking but logical thinking. the generating

of precise ideas, the combination of them, the relating of ideas inthe forms of judgments and propositions, the uniting of propositionsto form doctrines for the enlightenment of the humah understand-ing and the guidance of human conduct. Every me knows or oughtto know, for the fact is sufficiently obvious, t above each ofthe great types of human activity there hovers a shining ideal ofexcellence--La muse, if you will, or guardian angel wooing andbeckoning us upward along the steep endless path toward per-fection. Now, what is the muse or the angel that lures and sus -

tains our efforts in logical thinking? The name of the muse isfamiliarit is Logical Rigor, the name of an austere goddess de-


manding, though never quite securing, absolute precision; demand-ing, though never quite securing, absolute clarity; demanding,though never quite securing, absolute cogency. Is this unattainableideal of absolute rigor a valuable one in the conduct of humanlife? The answer, which cannot be too emphatic, is this In everydepartment of life where thought is required, the standard of logicalrigor is so valuable that just in so far as our thinking departs fromit, our discourse sinks down toward the level of mere chattering--the noise and gabble of our prehuman and subhuman ancestors.Why is it that the standard of logical rectitude is so clearlyu.,vealed in mathematics? And why is it that mathematics succeedsso famously in approximating conformation to time standard? Thesecret lies in the method of mathematics- -time method of carefullyselected and clearly enunciated postulates, of sharply and com-pletely defined concepts, and of painstaking deductions or demon-strations. Because there is no field in which a worker can escapethe necessity of making conscious or unconscious use of postulates,nor the necessity of formulating definitions and of attempting de-ductions and demonstrations, it is evident that mathematicalprocedure furnishes a model for the guidance of criticism of alldiscourse of reason, no matter what the subject or field to whichthe discourse pertains.

Mathematics and the Universal Concerns of Man, Timeconsiderations thus far advanced, though fundamental and decisiveregarding time title of mathematics to be listed among time humani-ties, are far from being all that might be adduced. One who open-mindedly contemplates the humanistic bearings of the subject willbe led sooner or later to see that, as 1 have said elsewhere,2 "Everymajor concern among the intellectual concerns of man is a concernof mathematics.'' No doubt that statement will seem to some tobe extravagant. Yet time statement is true and the truth of itought to be made known to all. Let me briefly submit a few justi-fying considerations.

Every one knows that among the most impressive facts of ourworld is time great fact of Change. Time universe of events, whethergreat or sniall, whether mental or physical, is an endlessly flowingstream. Transformation, slow or swift, visible or invisible, i3perpetual on every hand. But events are interdependent, so thatchange in one thing or place or time produces changes in other1pde l'hilwinphy and Other Ensayn, E. 1'. 1Mttun and Company.


things and places and times, With the processes of change everyhuman being moron, medioce, or geniusmust deal constantlyor perish. The processes of change are not haphazard or chaotic,they are lawful. To deal with them successfully, which is a majorconcern of man, it is necessary to know their laws, To discoverthe laws of change is the aim of science, In this enterprise ofscience the id'a1 prototype is mathematics, for mathematics con-sists mainly in the study of functions, and the study of functionsis the study of the ways in which changes in one or more thingsproduce changes in others.

We are here in the presence of another bearing of mathematicsupon a major concern of man. I mean our human concern toascertain what things, if any, are permanent in the midst of change.Human beings desire to know what things, if any, abide. Wewish to know what things, if any, may be counted upon. In thisgreat quest of permanence in the midst of mutation is found theunity of science, philosophy, art, and religion, for it is the sov-ereign concern of them all. Is it a concern of mathematics? Tofind the answer one has only to glance at the immense mate-matical literature embodying the truly colossal doctrine of In-variance.

Next consider the great subject of Relations. Such terms asspouse, husband, wife, father, mother, parent, child, king, subject,president, citizen, partner, enemy, friend, greater, less, better, worse,and so on and on, are familiar examples of relations. Whoeverexamines the matter will be astonished to find that most of thewords in any language, either directly or indireetly, either ex-plicitly or implicitly, denote relations. Each thing in the worldhas named or unnamed relations to everything else. Relations areinfinite in number an in kind. Being itself, said Lotze, consistsin relations. Science, said. Henri Poinear6, cannot know "things"but only "relations." To be is to be related. To understand isto understand relations. To have knowledge is to have knowledgeof relations. It is evident that the understanding of relations, thegaining of relation-knowledge. is a major concern of all men andwomen, whether they arc aware of it or not. Are relations a con-cern of mathematics? They are so much its concern that ablecritics have thought it possible to regard mathematics as havingrelations as its sole concern.

I have thus far said nothing explicitly regarding morals and


religion. Undoubtedly these are among the major concerns of man.A word regarding ethics. We know that, in any fairly stablecommunity, no matter how primitive, no matter how civilized,there gradually rise and ultimately prevail certain sentiments re-garding "right" and "wrong," regarding "good" and "bad," regard-ing what "ought" to be, and "ought" not to be, in human conduct.These sentiments get themselves expressed in the forms of maximsor propositions. The propositions are regarded by all, or by most,members of the community in question as embodiments of ethicalknowledge or truth. The body of propositions is an ethical systemgrown out of experience. Such systems vary from community tocommunity, and from period to period in the life of a given com-munity. What service can mathematics render in connection withsuch a system of ethics? The question has been well answeredby Jacques Rueff in his excellent little book, Des SciencesPhysiques aux Sciences Morales. He has here shown that, in thecase of any such system, it is always possible to find a set of prin-ciples or postulates from which the propositions of the system canbe deduced as consequences in the mathematical manner. In thisway an ethical system rises from the level of experience to that ofa logically or,.,4mzed doctrine. The transformation is one im-periously demzinded by the human intellect. Moreover. by sucha transformation an ethical system is shaped for criticism. Andthis is well, for, as Cousin long ago said, "La critique est la vie dela science." I should add that Rueff's book has been translatedinto English and published by The Johns Hopkins University Pressunder the title, From the Physical to the Social Sciences.

The bearings of mathematics upon religion are treated byanother essay in this volume. I will, therefore, content myselfwith a single relevant observation. It is that the concept of in-finity which is involved in the great question of limn; rtality, isdealt with in mathematics, but nut elsewhere, in strict accord withthe standard of logical rectitude.

In the light of what has been said and suggested in the fore-going discussion it is abundantly evident, I believe, that, amongthe agencies for qualifying human individuals -to create a goodlife on this planet by the use of human faculties.' or to representworthily, in their life and work, the great potential dignity of Man,Mathematics is unsurpassed.

MATHEMATICS AND RELIGIONBY DAVID EUGENE SMITH

Teachers College, Columbia University, New York City

The Bonds Between Them. It is one of the tendencies ofthe mind to look upon its own major interest as the focus of allknowledge. The mina tends to see analogies that are, at best,remote.; to magnify the influence which its own favored scienceexerts upon all other branches of knowledge; and to feel that itdetects bonds which do not exist. The poet, for example, seesin the Book of Genesis a magnificent prose poem, the upliftingpower of which vanishes when he thinks of it as a treatise onnatural science. The mystic sees in the Old Testament a field ofwhat lie feels is mysticism, and he reads into it a harmless ob-scurity that pleases him and has the merit of injuring no one else.The Christian apologist finds in its inaccuracies, as that 7E (pi)equals 3, the errors of sonic ancient copyist, instead of franklyrecognizing that the one who wrote that particular verse simplyused the everyday common value adopted by the people of histime. The mathematician may, for a similar reason, tend toexaggerate remote analogies and to assert a closeness of relation-ship between his own field of interest and that of the theologianthat. is, in fact, very attenuated. In speaking of this relationship,therefore, one must always be on his guard against trying to see theinvisible or to imagine that which has no existence. So much forthe initial objection of those who look upon knowledge as madeup of separate and distinct domains.

The bonds uniting mathematics and religion have often beenconsidered, and many have been the monographs written and thewords spoken upon the subject. The trouble is that the attemptshave generally concerned the theologian on the one side and themystic with some knowledge of elementary mathematics on theother. They have only rarely been made by either the seeker afterthe good, the true, and the beautiful in religion, on the one side,or the constructive mathematicio!, cf genius on the other. Even

53

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MATIIENIATIC:4 AND RELIGION 55

The Infinite. First, mathematics soon leads us to a feelingthat the Infinite exists. The inquisitive child shows this when lieasks his teacher what is the largest number; and the teacher showsit in her inability to reply. This feeling grows more impressive whenthe child becomes the youth and studies any elementary series,even the summation of ri terms of the geometric or any other type.It increases when lie studies geometry and wonders what happensto the sum of the angles of a triangle when the vertex is "carriedto infinity," and when lie asks the teacher what infinity means.It inereases when lie' studiei simple trigonometry and finds that,as an angle approaches 90' the tangent approaches infinity, sud-denly becoming minus infinity when it passes through the rightangle. It when, if ever, lie becomes a scholar in mathe-matics, and deals the infinities of higher orders, with trans-finite numbers. with the ,,veral plans of representing infinitygraphically, and with the infinity of time and space, or with thefiniteness of each. And finally. when he measures the known uni-verse, or universe of universes. and thinks in light years (thedistance that light travels in a year), and finds that the distanceacross explored space may be 400.000,000 of these light years, andlets his imagination carry him to the verge of this space and leadshim to wonder about that which lies beyondthen the mysterybecomes overpowering. lie has pushed back the clouds of igno-rance only to see that his own ignorance has become more andmore ho!leless, and that science leaves him helpless in the presenceof a new infinity. The childish boast that we will believe onlywhat we see, the most childish of all our feeble assertions of ourfaith in our puerile strength, avails us not. Mathematics haslured us on, and at the last we feel more helpless than ever, be-cause we have come to see how full of awe we are in the presenceof the awful Infinite.

The Changing Bases. Again, the yorth need not even reachthe legal age of the adult before allot hal, fluid of mysteries tendsto engulf hint. Ile is taught in mathematics that certain postu-lates are ,acred and that lie must not question them. In religiousinstruction lie is taught the saute. Iii mathematics he will be en-couraged by any honest and capable teacher to see that certainpostulates mu e. not always true: in religion lie may be condemnedif he queries certain others. In general, many teachers ill eachdomain display a kind of fear of lamest inquiry, a fear based either


upon ignorance, or upon faith in tradition. No field of mathe-matics need fear searching inquiry, and no religion or sect needfear the scientific study of its essential nature, however much itmay fear a study of the nonessentials which have accumulatedthrough the ages. In mathematics it is evident that each of theseoeries has the same number of terms, however far we go:

.

1 2 3 4 5 6 7 8 9 10 .2 4 6 8 10 12 14 16 18 20 .

for each term of the second is formed from the term of the firstthat lies just above it. If, therefore, the number of terms of eachis unlimited, the number in each case may be said to be the same.But the second series is a part of the first, consisting of every otherterm. Hence, in this case, the part is equal to the whole. Theillustration is a common one, and equm my common is the one whichshows that the infinity of points in one line is the same as thatin a line twi,.e as long, or half as long, or one-tenth as long, or amillion times as long. Therefore the youth in school readily colliesto the stage at which he sees that postulates that are valid fortime small field in which he has lived, and that are necessary insuch a domain, cease to be so when he faces the Infinite. A postu-late is an assumption of validity in sonic special region; but weoutgrow postulates as we outgrow clothes, whether in mathematics,in physics, in behavior, or in any other domain, substituting newones which appear to have validity in the new region of thoughtin which we find ourselves. When Einstein made known his theory,it did not destroy Newton's postulates in gravitation, or his laws.These are valid up to a certain point, or at least are practicallyworkable. Ile simply took the next step. When Lobachevskyand liolyai proelaimed their theory of parallels, they did not de-stroy uclid's postulames; they simply assumed another set andworked out a new lot of conclusions. In ordinary finite space,Euclid's geometry is a workable one; in spare in general the otherhas advantages. Euclid tacitly assumed that spare was every-where alike and that a straight line, however far produced, neverreturned into itself; modern writers tend to assume that space iscurved and that what we think of as :1 straight line is like a greatcircle on a sphere, always returning into itsrlf. Mat ...maticssimply says, -II this, then that"; it does not say, "This is eternallytrue, therefore that is eternally true.'' It never fears to have a

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L9 GNIV SDILVIV.M.LVIV.


Is it through a fourth dimension, which scientists now come to con-sider a commonplace idea? If so, what other spaces are there andwhat is their nature? Not without value are the rather childishconsiderations of Flatland and its inhabitants, and of the way inwhich a four-dimensional being may look at us and at the liveswe live. If this raises the question of other spaces, of other uni-verses of universes, the speculation, immature though it may be.has value. It places religion in a new light, it sets new bounds tohuman impotenee, and it takes away the boastfulness of a mindcharacterized by "arrested development." The same influencecomes with respect to time. Is it, like space, a closed affair, re-turning into itself? Is it. like length, a dimensionthe fourthdimension? We can point to the north. but can we point toto-morrow? Time has llw elements of a dimension but we are toothree-dimensional-minded to point to time direction it takes.

Algebra, like geometry, leads us to similar speculations. Theequation 2x + 3y == 6 is represented geometrically by a straightline in a flat surface (a space of two dimensions) : 2x + 3y + 2 = 6is represented geometrically )y a plane in our space of three dimen-sions: but what about the equation 2x + 3y + z + 4w = 6? Haveour dimension:- given out? Should it be a solid in a space of fourdimensions? And if so, where shall we end in our speculations?Into what kind of a super-cosmos are we being led? People say."1 will not believe in God," but they believe firmly in Nature,and most of them have the faint remnants of a belief in signs, inomens, in luck. and in looking at the new moon over the rightshonhiel. They will not believe in any possibility of a worldbeyond. but they will see the entire possibility. and at present theybelieve in the probability. of a dimension beyond our own. It is acurious situation, this religious skepticism. and it would scent thatit would tend to vanish if the theologian were not afraid of honestsearch after the fundamentals of religion. This. at any rate, hasbeen the result in the fields of mathematics and of science.

Obscurity of Language. Much of mathematics and also ofreligion is obscured by the language used. Mathematicians in thesixteenth century spoke of negative numbers as "fictitious" hut assoon as it was found that they could be represented graphicallyand used physically. they ceased to be such. They then heeameno more fictitious than a fraction. for we can neither look out ofa window of a time nor hook out of it, 2 times. Each number

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that quartz always crystallizes as a hexagonal prism joined to apyramid and that the bee's cell patterns after it, and has clone sofor millions of years and will do so for billions more, With theunseen tendrils of his mind he grasps the Eternal. No book, nopriest, no teacher, no authority has led him to see that the certain-ties of mathematics ha\ e helped to conquer what he considered hiscertainties of childhood. The essential features of religion offerno difficulties that differ greatly from those which he may easilyconquer in the domain of the Mother of Sciences.

Do children get this from their mathematics? Not when ledby teacher;; %vhose minds and interests have never seen the light-and so with their courses in education, in science, in history, inreligion, and in art. But a new era in the teaching of mathematiesis dawning, an era in which tradition gives way to a nobler concep-tion of what all +he sciences mean in relation to one another andin relation to the higher ideals of humanitythe fine arts, thecultivation of a taste for better literature. the social life and thecomforts of the world, and the religious instincts of our race.

THE :MAT.11EXIATICS OF INVESTAI..!,NTro- WILLI:1'M I,. Il.kIZT

Lan of Minnesota, Minneapolis, Minnesota

PAIIT 1. IsmobtarrioN

Extent of the Discussion. Along the backbone of the busi-ness vold we find those problems which involve simple interest,compound interest, and the applications of compound interest intl:e theory of both aniatities certain and contingent annuities. Theword annuity lucre refers to any sequence of periodic payments.The theory of interest and annuities and their applications is re-ferred to as the In athrrnaties of investment, or the mathematicsof finance; it is a part of the more extended field of actuarial sci-ence. The mathematics of investment, as distinct from mere arith-metic, is at the foundation of scientific banking. accounting, bondpractice, all forms of artivity involving the investment of moneyand the discharge of debts 1411:Ocularly the discharge of debts bysequenres of periodic payments, life insurance, and life annuities.In the present chapter, wi shall discuss the theory and applicationsof simple interest, compound interest, and annuities certain. Weshall not consider the theory of contingent annuities, and theirapplications, whieh occur mainly in a treatment. of life annuitiesand of life insurance. This aspect of the mathematics of invest-ment is not of such general appeal as is the rest of the subjert, and,moreover, its treatment presents essential theoretical aml notationaldifficulties.

Mathematical Prerequisites. A minimum satisfactory at he-

mat ic:al background for the study of the elements of themathematics of investment consists of arithmetic, and one and one-half years of high school algela a. Familiarity with the computa-tional aspects of logarithms is very desirable, but is by no meansan essential part of this background; it is merely convenient tosimplify the arithmetic involved by using logarithms, or. better,by using a computing machine. In any rational treatment of t he

In:ahem:It les of investment which does not introduce artificial61

(12 THE SIXTH YEARBOOK

mathematical difficulties merely as theoretical toys, there is nonerd of the use of logarithms in solving exponential equations,Or other advanced topics in the theory of logarithms. In thepresent chapter, logarithms will be referred to only in brief dis-cussions where an effort will he made to properly gauge their use-fulness. or the necessity for their presence. Knowledge of theformula for the sum of a geometrical progression i the only itemof the theory of progressions which will be needed, and this for-mula will he used only once in the chapter. This rather extendeddiscussion of the rifle which logarithms and progressions will playhas been given in order to dispel at once an all-too-prevalent no-tion that the mathematics of investment is heavily dependent. onprogressions and logarithms. A proper foundation for the mathe-matir of investment consists mainly of good arithmetical skill.that familiarity with the use of liter:II numbers and algebraicmanipulation which results from one a I one-half years of algebra,and stud' maturity of experience as 1S possessed by students of thesenior high school, or higher levels.

Historical Background. The mathematics of investmentdeveloped very (ally, along with the use of interest in financialtransactions. From the year 1202 onward. books on mathematicsincluded prob. 'mss concerning both simple interest and compoundinterest. At an extremely early stage 11 e. find problems of greatdifficulty, as juligod from the standpoint of the early mathema-tician who Inn! neiti,rr the theory of logarithms nor the extensivemodern interest tables nt his disposal. Thus, Fibonacci (12021proposes the following problem:'

A certain man puts one denariu at- interest at suril a rate that. in fiveyear.: It: has two denarii and in every five years thereafter the moneydoubles. I ask how many denarii he would gain tita this one denarius iti100 years.

In Tart a uwral Trattuto (155(1) . we meet the follow-ing problem2 whose solution by 'rant aglia, with his inefficientmat hemat lea! tools. demanded great- ingenuity:

A merchant ga%. univ..rsity 2.81-1 ducat:i on the understanding that heW:IF to pay 615 ducats a yeLr for nine yea,.s, at the end of which the 2 814clueats should he considered as paid. What interest was he getting on hisilioncy?

Cora Sanfurtl. .1 Short tory t,l Abfilic»m jem,Compn nY. 1 ti:: 11.

V,rit zsattford. op.

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any treatment of it at the secondary level, the discus ion of thepresent chapter will presuppose no knowledge of the subject out-side of the familiar facts about simple interest and compound in-terest.

Outline of the Discussion. Part 11 o: the chapter will bedevoted to a discussion of certain aspects of simple interest, andof the related topic of simple discount. Admitting that the mate-rial referred to in Part II is present in high school arithmetic,Part II aims to orient this elementary section of the subject withrespect to the more advanced portions. The author believes thatsuch orientation could with advantage be emphasized in the teach-ing of arithmetic.

Part III will he devoted to compound interest, with emphasison simplification of method. Such simplification comes from (1)the use of the interest period instead of the year as the funda-mental unit of time in the formulas, and (2) systematic use ofinterest table: and interpolation in them, in place of logarithmicmethods.

Part. IV will be concerned with a derivation of the funda-mental annuity formulas, and their application in various typesof problems. The length of Part IV, relative to Parts II and III,does not properly indicate the flitstanding importance of Part IVin the mathematics of investment. Lark of space for descriptivematerial makes it impossible to present. more than a brief indi-cation of some of the many applications of annniTies certain.

PART II. SIMPLE INTEREST AND SIMPLE DISCOUNT

An Algebraic Treatment of Simple Interest. At anyschool level beyond that on which the student has dealt with linearequations, the treatment of simple interest and its applicationsshould be definitely tied to the familiar equations:

/ = Prt; (1)S P -4,- /; (2)S= P(1 rt). (3)

In these equations, P is the original principal, r is the interestrate, expressed as a decimal. t is the time of the investment, ex-pressed in years, I is the imerest earned by P in t years, and 8 isthe amount due at the end of t years. We should not confuse the

MATHEMATICS OF INVESTMENT 65

student byh long- winded word-descriptions of how to find the ratewhen the principal, the interest, and the time are given, or of howto find the principal Nam the amount, the rate, and the time aregiven. Ile should be taught. to substitute given quantities in prop-erly st,leeted equations, anti then to find the uuknowns by the nat-ural algebraic procedure.

In a treatment of the mathematics of investment, simple in-terest is important not only on its own merits but also as a meansof presenting the important terminology of present value, amount,accumulation, and discount under a simple guise. In equation3, we call P the present value of the amount The possession ofP to-day is just as desirable as the possession of S at the end of tyears. With account taken of the effects of interest, P and S areequirah nt sums of money due on different dates.

To ace/cm/date a principal P for t years at the rate r means tofind the amount S due at the end of t years if Pis invested at therate r. To discount an amount S for t years means to find thepresent. value of S on a day t years before it is due. The discounton S is the difference between S and its present value 1', or is(SP).

EXAMPLE 1. (a) Discount $1,150 for 21,:i years at the rate simpleinterest. (1) Find the discount on the $1,150.

Satati,,o. (a) We desire to find the present value of the amount S $1,150,212 years before it is duo. We uso equation 3 with L = S= 1,150, andr :

1,150= P[1 +

1'[2 + 5(.055)]2Z5P=--,

P

2,300.

2.300

2,300z.275

$1,010.90.

thi prcscnt aluc of 81.150 is $1,010.99, the: discount is 0,1501.010.90), or 5139.01.

In the preceding example, 5139.01 is the discount on $1,150,due at the end of years, awl. also, 5139.01 s the interest for21,:, years on the principal $1.010.99. That t, 5139.01 plays adouble role.

The last paragraph illustrates an if.teri sting fart. From equa-tion 2, 1 11_Nice, the couti,tity 1, wi,ih w dolinerlas the ihterr.it ail the P, has inAv been given a second

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-apui ti o:1...,1 (31 svmq.;xo3 ;0 .133;p:; 41:do13,1,1; ,(q2 .ill((

rem .mpao ul Ytpuntn 9 jo ptn otp 11! Svci 0.1 041" I 111"I'' lmiA1 (v) DativApp ul 01(1) :.i):(1 9:,9 Forfn v f ;41.11v..T

9 pul; i.:(LO! 41:111)a JO osti .cq pair)

-um{ ci 10:,) .0010?A1o: 111 airptc.1:(I 14,)a.)411! jn (autiN aill 41:11.\\ ) 111110,10 ;:tioililatplcill

ail pm? so 11111 ..11)-:-:11T10.1(1 in :Fitil1t1ll0.411) mil 111 si11,)1(10.1(1 11:11:411

Jo !IN- .)Intoos!i) ao1 poi: ),:loitt! oplitt14 anj tilog T

011ITIOp Op 1: .1 a.)p?Ol ( )1 liittilo: loll op 1)1111 '9 Dell: '9 't, suo!iiho jO aoul illoptwdopo! aiii ppi.oaci

`7uno,).11) R/t/n/ti aa/t/m/)// /t/ 1dadiu.i oiduqs c,zpOlooaa amio paom aril a:411.)i; Samilp.10 otil

111 `)Nad3)11/ lou pur poo.w//, st 1otioot-:!1) astvuaaq -anio(1 41111.\\outoi: 0,)1/1).1/),/ /i/ alqrnind a1d1111 i.311

()I poa.m.loa 111-InIa3 st 1,1tio.vi1) aplin!R 41111.11-1,3 LIT

"07(.16tS, 091. Onf: ef '(..? Unlit:11W Tuw:I 'OTI )(90')OOR `t uolvonba mom !co p :00E Au: am floyiniog

dro.i: mil ni s:1:p ofr, niriuqs 049 ')TT

quogo.iti SII 'i:.01) 06 Jo ptio alp p: imp s! onc:.s. 7, TI<Mvx:.1

11111! :411op1 :/11)) 311 4.1001111! slz 111r Anz 9 put: ,g :4110(.111111Y

(9) .(ip .1); 7= CI

(g) !/ --r/ () T

JO) stinpimha 1.\\01 in; otp () I pi am om

--- I /PS' I

ips / tiatu, Totilaap 11 ill pas:za,i(Ixa

'IT 41. 111%) '4uno.):411) aidill!A Jo op:a 11 oonpoalui 'O..: op 0,1,

lunoun: oft) 10 411J. 3.`dilltioaaad I i4:43.1(1X0 10a1.111).%

-110a S.Ta.% 1.0 4! 11100011: alp 110 11010,W11) Jill '.thitaapistioa

II! 11411ot/tad alp Jo aroS. Jac! ath:111oalo(1 TIT1I4.IJJ

goi:gagixa 11;)111.\\ m11.1111141 alp 0s11 0.1x ',/ 110 isaaolu! alp sI1

04 III '110c1011 luo)aoclitii III: 04 .iivatilint -tomb atil lio!sstiostp :Fiti!paaaad alj.I unoosTa aicitu!s

019,a omnop sSuid T .141 pm() um mil up iuttodNip .)1j4 osp? alum

NO0 TUX 114


I may receive $1,000 from him now ar; the proceeds of my loan? (b) Whatinterest rate ma I actually paying?

Solution. (a) Interest in :ulvanee, at the rate 6%, means that simplediscount is being charged at the rate 6%. In equations 4, 5, and 6, we aregiven 1' = 1,000, t and d = .06. We use equation 6;

1,000= 1/2(.06)1;.97 S = 1,000;

000S = 1

.$1,030.93.

97

(b) From Part a, I pay $30.93 interest at the end of 6 months in additionto the $1,000 which I received at the beginning of the transaction. FrontI = At, with P 1,000, 1= 30.93, and t == 14,

30.93 = 1,000(r)(1;2),On solving, we obtain .0619. Hence, a rate of 6%, payahte in advance,on a 6-molt h loan, is equivalent to a charge of simple interest at the rate6.19%.

The terminology of present value, amount, accumulation. anddiscount applies equally well both to simple discount and to simpleinterest di-...ussions. To discount an amount S, due at the end of tyears, under simple int(rest at the rate r, means to find P by useof S= P (1 -I- rt). To discount an amount 5, due at the end of tyears, under simple discount at the rate d, means to find P by useof P = so. dh. The equations of simple discount. are easierto apply than those of simple interest, if we desire to find a prin-cipal P when the amount is given, that is, when we desire to dis-count an amount S. The equations of simple interest. are easierto apply than those of simple discount when we desire to find Swhen P is given!'

PART III. COMPOUND INTEREST

Definition of Compound Interest. If, at stated intervalsduring the term of an investment, the interest due is added to theprincipal and thereafter earns interest, the sum by which the Origi-nal principal has increased by the end of the term of tla. invest-ment is called corn ponnd interest. At the end of the term, thetotal amount due, Which consists of the original principal plus thecompound interest. is called the compound anzmud.

Hereafter, the unqualified word Liz/crest will always refer tocompound interest.

tir Iii'11-,1"11 of I nlations sImpli. simple(11111.t..r 1 'if tho authur's .11,11ht ,,111,rx of Inrixlistr sit, RI rims!, 1). C.

litlith and I %(Itupitily,


We speak of interest being compounded, or converted into prin-cipal. The tittle between successive conversions of interest intoprincipal is called the convergion period, or the interest period.

Thus, if the rate is (31/4, compounded quarterly, the conversionperiod is 3 months, and interest is earned at the rate 6`10 per yearduring each period, or at the rate 1.5',1e. per conversion period.

ExAmpLE 1. Find the colu;.ound amount at the end of 1 year if $100 isinvested at the rate 8%, compounded quarterly.

Solution. The rate per conversion period is .02. The original principalis $100. At the end of 3 months, $2.00 inter( st due; the new principal is$102. At the end of 0 the intt rt,t due is 2r/e of $102, or $2.01,and the new principal is ($102 +$2.041, or 814)1.01. In this fashion, we findthat the minutia at the end of 1 year $108.213. The total compoundinterest eant:d in the year is $8.213. The rate at wide!) the uri).:inal principal

. 8.2.13increased during the year is or 8.213%.100

,

Nominal and Effective Rates. In the usual way of de-scribing a given variety of compound interest, the rate specified iscalled the nominal rate. It is the rate per year at which interest isearned during each conversion period. The (IP etipc rate is therate per year at which interest is earned during each year. Thus,in the preceding example, the nominal rate is 8(7(..; the eeetiverate is 8.243,-; ; the rate per conversion period is All of thesethree rates of interest should lie kept in mind when considering agiven variety of compound int(.'rest.

It is emphasized that, in the future. when we refer to a rate ofinterest, we mean a rate per ycur, except when we otherwise spe-cify a rate per conversion pi riod.

The Compound Interest Formula. Let the interest rate perconversion period be r, expressed as a decimal. Let P be the origi-nal principal, and let . be the compound amount to P ac-cumulates by the end of k conversion peritols. Then, we shall provethat

S=1)(1+ rik. (1)Proof. The original princilial invested i P.Thu interest due at the end of 1st period Pr.New principal at the end of 1st period is /' Jr ' -1- r.Interest due at the end of 2d period N r pi 1 I, or Pe (1 +r1.New principal at tit(' 011(I 4)i 2.1, period is

Pt], r ) -; Pt 1 + r)r n + r), or P11 r

MATI EMATICS OF INVESTMENT 69

By the end of each period, the principal on hand at the begin-ning of the period huts been multiplied by (1 + rl. Hence, by theend of k periods, Lie original principal P has been multiplied ksuccessive times by (1 r), or by (1 + r)k. Therefore, the com-pound amount at the end of L. periods is P(1 + rik, as in equation 1.

The exceedingly elementary discussion leading to equation 1

was not given with the idea that any new facts were being pre-sented. The object of this discussion was to exhibit the convenienceof the terminology of conversion periods, and rates per period, andto show how this terminology Icads to a simple general formula forthe compound amount.

As in Part II, we call P, in equation 1, the present value of theamount TO accumulate P for k vonversion periods at compoundintecst. means to 1111(1 the amount S by 1151' of equation 1.

Ex.\ m Ns 2. Accumulate $300 fur 9t years at 6%, compounded quarterly.Soiat;on. The rate per conversion period is .06 4, or r .015. 'I'll(:

number of conversion periods is 1: 1. 37. From equation 1,300( 1.015 )" 3(0 ( 1.7:j18 ) _ : $520.44.

In the solution, we obtained the value of (1.015)47 from Table I, on page Si.

If Table I W01(' 110t :I:Vadat/IC, 11.015)37 would have to he1w use of logarithms, or, less conveniently, by use of the

binomial theorem. For this reason, one might rashly say thatlogarithms were essemial tools for further work. This statementis unjustified, because fairly complete tables° of compound amountsare just as accessible as tables of logarithms. The whole discus-sion front here on presupposes that sonte convenient set of interesttables is available. For the illustrations 41f the present chapter,the accompanying extracts of tables are sufficient.

To discouitt 0.11 0.11101111t S for conversion periods means tofind the present value of .S on a day k perioils before is due. Tofind I', the present value, we solve equation I for I' in ternoi ofwe find P (1 + ki Or

P (2)eciwition:,. I and 2 ory Notivalma tNe refer to

ti,cto as t1:(' fundatneotal (foal irats (t1 coMpollndExAmPLK Du-rqvint fur 112 YP:11.6 , compounded sem:annually.Sotio'ion. Iii f fii1.1(ion 2, \\ hate $50, r ", .02, and k 2Ei1)

:7 50(1.02, 50(.5:)76, $41_81.

The di..vouto on the $50 is (50 -11.S11, or $8.16.see. 1..1- !, Hie nut ldrs from lb, furtxtmnt,

1,. C. 11-.011 1t1;.1:0.

70 THE 6IXTII YEARBOOK

Comment on the .Fundamental Formulas. Recall that, inequations I and 2, the unit of time is the ronversion period, andnot one !par; the symbol r is the rate per conversion period;is the time, expressed ill conversion periods. This feature resultsin a worthwhile simplification. It ve bad let n lie the number ofFears for which P is invested. in be the number of times thatinterest is coverted in each year, and j be the nominal rate, thenwe would have fianid that

-= / ( 1 + (3)

The greater complication of 3 ) as. compared with ( 1 ), is thereason why we used the conversion period instead of the year asthe fundamental unit of time for the deseription of data. We donot employ equation 3. :11(1 recommend its complete submersion.

Compound Interest for Fractional Periods. 'Flue funda-ental definition of compound interest gives no meaning to the

notion unless the time of the investment i5 au integral number ofconversion periods. That is, up to the present. we hate aSst111101that. k, in equation l, is an integer. If the time of the investment isnot a whole ttilitilwr of conversion periods, it is customary hipractice to define the compound amount to la. the result obtained asfollows:

( 1) Find th«.ompound amount at the end of the lust wholeconversion pr rind contained in the giien time.

(2) Accumulate the (sulting amount for the remainder of thetime at simply.' interest at tier' gir( // nominal rate.

EXAMPLE 1. Faill Thi :lamina at tilt` of 2 years and 7 months, if81,0(.X) inxIsted at 8`;

Su/Wiwt. 1:t -1 int orost (.1:tt 11e. tern' of the investment. is at theend of 21..2 pous, Tli, la the ent1 of 21:s years is. by equation 1,

$LOW t 1 .02)' 1,(X)011 .2190 $1.219.00.'rho To111:1 Min.: tin. 1 month. Iienee, neiannlitt... Ile. new principal,$1,219, fur I it ,..1111110 111 1. 0 I. We 11,4 i'rtNvoli P *1.219, awl 1 1'!3:

I 1219 08)1.-11!) $8.13.

hv ntilmint at t nci of 2 y,--trs -Ind 7 months is ($1,219 -1- $8.13), orS1,227.13.

In lilacs of the definitiioi of ti.c eninpound amount for a frac-tional period cci,it it \y have just used. it is customary to agree,


theoretival work. that. cren whrn 1 is not an integer, the pres-ent value P and the amount S shall he related by equation 1.

For contrast. the definition used in Example 4 may be called tliepratica/ definition of the amount for a fractional period. and thesecond definition may be called the themTtirai definition. Thezonount given by the practical definition (litters only s'ightly, inthe usual problem. from the amount. given by the theoretical defi-nition.

Unknown Rates and Times are conveniently found by simpleinterpolation in compound interest tables. The process of inter-polation referred to is the same iLs that employed in the use oftables of logarithms or tables of the trigonometric functions,

xAm Pix 5. If interest is compounded semiannually, find the nominalnor at which $1.000 accumulates to 81.200 in 51:2 yoars.

1, r the unknown rate per conversion period. We haveI' $1,000, S $1,200. 1 I: 1am. 11. :rum equation 1,

1,200 = 1,000(1 4- r)",(1 + r)".---r 1200.

In the row of Table I fur k 11, we find (1.015)" 1.1779, which is lessthan 1,200, and 11.021" .-.1.2431, which is moo- than 1.200. Heuer, r is be-

.015 and .02. In finding r by interpolation, we MI--(1 +that tho same proportion of the way from ----;015 io .02 us 1.200 is of the way from 1.1779 to 12434. 1 .. '015 1.1779_

1.2134 1.1779 .0655; r - ? 1.2000

1.1779 -- .0221. 1 .02 1 1.2434

.02'21I

tleure,r is f the way from .015 to .02. Since (.02 .015) =, .005,.0655

r .015 + 221--( 005) _015 + .0017 .0167.655The nominal rate is 2r, or .0334, ur alTroximatily 3.3":'E. This final resultis altoo-4 certainly aceurat to tenths of :t per cent. .1 ,:erund approximationcould bo found by inti-rpolation, by use of cou....lrable computation' but.the present result would satisfy most. petical

xAmpt..; 6. IIow long will it take for $5,250 to avettinulate to $7,375 ifthe money is in% eNted at GYE, compounded quirtrly'!

Sobetion. Let I: n 1 ht. Iiici 11111111,1* Of conversion l ''iiutls. Fromequation 1,

7,375= 5,250(1.015)k;

(1.015)1' =;)

= 1.4018.,2a0

By interpolation in the column of Table I, we find that k 22.83,periods of $ months. The time requird 11(22.83) years, or 5.71 yours.

Ise/. cuattn. tit 1. p. 3 I, iti liorS Mit/h, /pa: irm of /tic, Nim,


There is ample justification, on the grounds of accuracy andsimplicity, for using a certain logarithmic method in Example 5.Ifowever, in the analogous and more important type of problemrelating to annuities, this logarithmic method is not applicable.For this reason, we adopt the interpolation method as our standardone for determining unknown rates.

In Example 6, we solved an exponential equation in k by useof interpolation. It can he proved a that the solution obtained byinterpolation is an exact solution of the problem, subject to thenatural limitations of the tables einplo:k ed, if the practical defini-tion of the previous section is adopted for the compound amountfor a fractional period. On tin other hand. a solution of the ex-ponential equation in Example 6 by use of the customary logarith-mic method gives an inaccurate result, according to practice asexemplified by the practical definition of Section 5. This is aninteresting point, in view of current impressions in iegard to theapproximate nature of results obtained by use of interpolationmethods.

Equations of Value. To compare two sets of financial obliga-tions involving sums of money due on various dates, we must firstreduce all sums inyolvez1 to equivalent sums due oil some commoncomparison date, An ((illation of cultic is an equation stating thatthe sum of the values, on a certain comparison date, of one setof obligations equals the sum of the values on this date of anotherset. Equations of value are powerful tools for solving problemsthroughout the mathematics of investment.

Ex xtpu.; W owes 1' (i) $100 due at the end of 10 years, and (b) $200duo at the end of 5 years with aveurnulated intere:q at the rate :3%, com-pounded W wi,hes to par in full by making two equal pay-ments at tilt' ends o: :id and the 61)1 yea,. II nioney 15 nOW COnsitleredworth ; 'r, compounded Mt niiannually the credo find the size of W'sequal payIllaltS.

Sulution, Lot S.( be thi paynpqa. i %%L.:1w: to replace his old obliga-tions by two new ones. iquation 1, Patt 111, obligation b requires thepayment of 200(1.0).5," :11 Iii'' -lid of u year.

hid 3b/;riations New Ob/iga tcu; $100 due at the end of 10 yr. $x due in 3 yr.(b) $200(1.015)" due at the end of 5 yr. $x due in 0 yr.

,hall uso the end of .5 y-IN as a t.,Inp:in7ni date. In the following(VW I, ill of till' I' it tut InbI j.` the :11111 Id the equivalent values of

w. I.. 11:1 rt. Ow tit <tit Matitcniatiral 31uth!y, Cvl. XXXVI (192U), p. :17U.


the old obligations at the end of 5 years, and the right member is the sumof the equivalent values of the new obligations at the end of 5 years. Inwriting the left tuttbr, v: dim.,,ant (a) for 5 years, and take (b) unchanged,beeaust. (h) is din at flit- ntl of 5 years. In writing the right member, we(111Umulate the ilia Sr for 2 years, itnd di8emint the second $x for 1 year.In diseounting. or ztecinindating, we use equation 2, or equation 1, as theease net.. ho.

100( 1.02) to 4. 200(1.015)10=z(1.02)4 x(1.02)4.100(.A203) -1- 200(1.1605) = x(1.0821)

31.1.13= x(1.0821

1 4 .

+ /4.9612).+ .9612) =

$153.72.

2.0136x.

x-23 .0 ;111

PAliT I V. A N N CERTAIN

Annuities. An annuity is a sequence of equal periodic pay-ments. An annuity ccrtain is one whose payments extend over afixed period of time; :t contingent annuity is one whose paymentsextend over a period of time the length of which depends on eventswhose dates of oreurrenee cannot he accurately foretold.

Thus, a sequene of (-mud payments made in purchasing ahouse on the installment plan forms ail annuity certain. Thepremiums on a life insuranre policy form a contingent annuityhermise the premiums cease at the death of the insured person.

NVe shall deal only with annuit jog v(rt :tin, and. in this discus-sion. the qualifying. word certain will he omitted. The sum of thepayments of an annuity which are made in one year is called theminim' rent. The time between stieNssive payment:. is called thepawn eat intrria1. The time het weer) the herrinninfr of the firstpayment interval and the end of the last one is called the term.of the annuity. Unless otherwise stated. the payments of anannuity are due :it the ends of the payment intervals; the firstpayment is due at the end of the first interval, and the last. pay-ment is due nt the end of the term Of the annuity.

Thus. in the case of an annuity of $150 per month for 15years. the payment interval is ni.e month, the annual rent is$18(10. and the term is 15 years.

Present Value and Amount of an Annuity. Under a specifiedrate of interest. the prsent ratite of an annuity is the sun) of thepresent values of ;ill the payments of the annuity. The amountof an annuity is the sum of the compound amounts whielt wouldbe on hand at the Pn,1 of the term of the annuity if all payments


should accumulate until then from the dates on which they aredue.

Illustration. Consider an annuity of $100, payable annuallyfor 5 years. Suppose that interest is at the rate 4q, compoundedannually. We obtain the present value A of this annuity by add-ing the 2d column of the following table, and the amount S byadding the 3d colunni. The values of the various powers of (LW)were taken from a table not included with this chapter.

PA N or SinnDE K A I. END ur l'ItENI:NT VALUE or PAYMENT

1 year2 years3 years4 years5 years

CoMPorNI) AMOUNT' AT ENO OP' I

TERM 11, PAY NIENT Is LEPT rAIELATE AT INTEItFHT

100(1.04)' 96.15.4 100(1.01)4=-- 116.986100(1.04)-1 = 92.156 100(1.04)2 = 112.486100(1.04) ' = 88.900 100(1.04)' = 108.160100(1.04)' 85.480 100(1.04) == 104.000100(1.04)-'=: 82.193 100 T..: 100.000

(add) A $445.183 (add) S ,-, $541.632

Annuity Formulas. In applications, the payment interval othe annuity involved is usually found to be the same as the con-version period of the interest rate. This is the only case which weshall treat')

Consider an annuity which pays $1 at the end of each interestperiod for n interest periods. Let (air, at i) represent the presentvalue, and (sq at i) represent the amount of this annuity when theinterest rate per conversion period is i.

To determine a formula for The first $1 payment is due atthe end of one interest period. and has the present value (1 +The present value of the second payment is (1 + i) etc., thepresent value of the next to the last payment, due at the end of

periods, is (1 + The present value of the lastpayment is (1 + The present value of the annuity is the sumof these present values, or

arc.= (1 + (.1 -T- h ++ (1+i) I. i) + (1i)-1. (I)On the right s;de of equation 1. we meet a geometfic progression,with the common ratio equal to It 1). The formula for the sumof a geometric progression is crl c r where r is theratio, 1 is the last term, and a is the first term. In equation 1, we

01.'nr a grh-rni tr.atmnt of all ,a-,os whioh ariSP to th.' arplivations of an-nuitiPs. sec, author's Mathematic:: of Inrcatment. chapter IV.

94c.093 tu!xo.tki

r 000.9s) :4! ....not( sitil Jo (,tlit:A Th.) situ, .0v.1411.01$ 1

"1"1.4 ) t )(001 17.0' ti: 210)(X)0.1

not ;

(11(1.11. ottpl.% 111.)solti otii JO )rtiv% Tu.) plop:At/Wu JILL

%mini splip.id Zi 19 1111,11 ilil .7.0 lit/poti 111)!:,1,1.1111),) 4.01 4:,:,1011.1! ;AD

:r111:,). 9 1.1 111.101 tisUq.11 . {limo(; Ill; (X)01 111)/0/LIS

OS111)11 ,,tit 301 0.)!.1,14 1114./ M,11!.1l111),)

-11.10,) 114.111m x1. ,co11i)111 jj s.11:o. 9 .1ij Sit Moll' o 11.)1.. 111 1111., oils 11: OUVIS

1.11U 11:411;) (XX).9:41 .c C.1 hiarn: i11:111 1: siznott C Ilillr,11/.).1116 III .1

Pt) c(litti 0111 04 110!4J111)0,1111! it( 1)1.1,11)1S110,) Oil 41)11 !Moll liapiA SIUJI(10.1(1 11! 4111.).)Xt) ptn: s1101 4 0100.1 oil 11 4I.11

4,1t).11 ()I 1)II1: 1)111: i, stlt)(11:10)k) 011,10111(015 .q,) 4.11diticw 114 ;.:11

:4.1\011 1; b,.)1(1114 .11Ilittuu 4ttollsistxst .1() dotw1s1s.0 suotpd4

till Ppm zsuituttauj mil Jo lit)! ii:0!1(.1(11: .111; 41:114 Tipp 04

Jilt) 11 UJ1 411nIttl Pm: (:, 51:111111,1u.; to 1101 }1:0Hilitio,) 0.1141:1Ja Jlly (V) =

Am' = I. JJLILli

satup 4utiouti: dip 'S.I.ItlIttlits Jo puolt: -tit tit! 4U) I. \1:11 itiva ciatim ottit:. 1110:4:11(.1 dto al 1.101.(1.11 CR) SJUII 4

(1/ Jt1 lum ltiosoad J114 Jo 4uno111u put?

J1111:.1 411J5J,11.1 J114 41.10S0.1(10.1 I' 0'1 111.10} J114 tii Jo adqiittat.

Jto i t U puu spui.taci ttat:4,1,1.%110,1 aoti ,)411.1 4so.1,)}111 ! '4uotti.ud ilputadd otil tit (11 tiotti.% tit .1!titzto; aJill....110,) ..wx

1 (.1 -f 1)

t ! -1- )(-- -- ( -

- ( / - r- 1

--- (./ d tip .111

1stud Jo 1 viittwoj .o.wpadif u auj zutopouttosiu up11(11) to:,) ,).%\ dip to) 111.1.)1 0111 111,11)4 11.1I11.11 1(.1sit 1

IC J114 JO 1)11J 011 01111:s jo) ti}1111()1111: .1114 JO llillti 01.11

21N. Put: .)1It1t1tt: dip sitwitt.0:i1. alp ju m1r.% .1111

Joins sdattoli 1111:111(1 ,).\\ plosJaci

ti 1.l }t :1lllllllJ,)l: 0.11 11:T1t 111:00.1 0.11 -aN .10). 1)//1/11.1411 lifq

(.1 -1-- 1) -I 41111

!MU 3.11 `),)I1li a (./ -4- 1) =: 1) (., 1) ') ! -1- 4) .t ;111211

(12, Ss)LISIVAILI,V1


EXAMPLE 2. If you deposit $50 at the end of each 6 months in a bankwhich credits interest semiannually at the rate 4c'e, how much will be to yourcredit at the end o' the 20th year?

So/ution. The amount to your credit tho amount S of the annuityformed by the de-msits. We use equation 5 with R = S50, n -=. 0, andi

S = 50(.47 it .02) -.= 50(60.4020) $3,020.10;in the solution, we used Table III.

ExNtrhe. 3. I purchase a house worth $12.000 cash. I pay 52,000 cash.I also agree to make equal payments at the end of each 3 months for 9 yearsto discharge tha balance, pvinc;pal and interest included. If interest is atthe rate 6c1-, compounded quarterly, what must I pay at the end of each 3months?

No/ut:mt. After paying 52.0(X) cash, the balance is 510,0(X). This $10,000is the present value of the annuity whieh I shall pay, quarterly, in dischargingtoy debt. Let 1/ be the unknown quarterly payment. Then, :ye use equation4 with :1 $10,000. R unknown. a 36, and i .ot5:

lopoo=-- Rtair at .01.5).10.000 10 000R

a Al,;]---= =--- $361.5:: (Table IV)(.,- to 27.6607

Amortization of a Debt. A (1(.1)t, N% hose present value is is:aid to be amorti:ol under a given rate of interest. it all liabilitiesas to principal anti interest :ire dischare...ed by a sequence of periodicpayments. \\ hen t1.- payments are egital, as is the case,they form an annuity w}:sn present value must equal ..1, the orig-inal liability. In the precettit. Example 3, we tietorntined thequarterly payment vhieli Wt111 a debt of $10.099 withinterest at 6'; , rompountled ( tt 1'7zr!2, in 9 years.

EX MPI.E farm is worth 5.25,(XX) cash. A buer will pa: 512.000 cash,and he will amortize the balance by payments of $1,000 at the end of each3 months for 3 years and 9 months. At what interest rate, payable (luarterly,ts the trAnsation being tewetited?

s,,/:i(ion. '1 h.. balance due. after the cash payment, is 513,000. Let t bethe iinknown trorsst ran., per vonversion pprimi tut 3 months). Then,5E1 00) must equal the present Va hit', at the rate i, of an annuity of $1.000p.Ivabl quarterly fur 3 ysirs and 9 months. We use fornotla 4, with a 15,

$1000, .4 $13.(Xx), anti i unknown:13.000 1,000u1,-; :It i): i) 13.

\V,' solve tl:e last etvatibn by interpolation in Table IV. We seefrom the table that at -: 13.343, and ta::. at 2`; I .7:1 2.S19. Hence, I is between 11 ; anti 2' liy a solution like thatof Example 5 of Part III, we finct that L: .0185: since this is the

&19 *11 'r4..u4p,...,11, /4 YqPn"."111'11 S.10101111 till 1 0.1: 11,,N,11.*:11, O1110,1 U 0.

IS0.1A.411! puu 11113111.1d s;ti tid. ..cpoinno, tilat)m SJUOUI IllotiloolZIV 0111 ti 1:m 0000113 :4Mti.l.luti V .9 uldKvx..1

1X311 0t {1 JO 1101 }1:1t):4 J:1 4

1110.I3 011 ,:111 1".:ol11t: 111J111

-S1?ti 111111.1 0111 1101 401%41111.11 0114 Js01,1 111..11 tia!t{.\\ }uotti.ird ,witizo.u!

iutql 0114 :atii1)1111 a,:.)114 iimp ritiputti -loop Jo cuoiLi,R1 loout 0.11 'JZIS 110t.ji,.)(IS .111)01.10ti

icI '00.101111 j0 041%1 110.1.1.33 11 411 021.001111: Jti 01 si JIIS poulootis

tt 31 .;uatuiSEd uopuzpaouiv IBuL4 e Jo uopuup.tua;aa

:;t:inut.t03 Fluouivpunj j0 , ut)priliti!tiutu

S.(11.:.-.111,1)111

-(10.1L1 ttotts 0.1 10S 0.11 :S1111111.10.1 .11011 1.1010.101I )011 01) 0.11 '0111)

S01 411111111: 30 ,z,41111011111 ,I0 's01111:,1 .411JS0.111 311} J11;1:1.13101) 01 .111.10 111

'54113111.11(1 J111 :;(1 1)3111.101 3111) ..11111.1111: 011 1 30 01111:.1. 4110:40.1t1 011 Si .13111

011 1 .10 011111.1 11S13J ail} 4nti1 pitwo )\ 01(111111X:I Ill '111.10

0111 it) 1)113 3111 3.10334 1)01.13ti 3110 011111.1:d 1:-111 311 11ut;

'111.131 S11 311 1-1111.11[11.113(1 0111 1U .11q,11:(i rI J111) ..1111111.11.: 111: 30 1u3111,041

15.111 JILL *;:11:.1.1J1111 4110111.il:(1 0:/4 311 ethl??//itri.)(/ J(14 p; .111,),p)

attl 3t 'alup tuoli papIlf at? '<PT iiMIU/(/) 111.: 111.101

1111:S Si S 4110111S111I 0I1)011J(i 113111)0 JO JJ110111.30S V Opp loam) 11 '

1101 }L' .110[1111 tmtunj s1uJ11t.i.u(1 alp

(AI °WI u!'11.)

'C',81341.13 =-- 0 = 00 Oi;

(191-91-t, + 1) Y = 000'0Z +1)il == 000'oz

(cio41: !Ivo?, + =oonz ') 10.1.111j.

nthnu.c.ed kfutu!v-tal my JO on11:. 111,1:40.111 J111 -znici 11,11,) 'll ,si .1111 s1at11).) .)stimi .1111 jc) .11111:% 311I. _!:1),11 r.t 1,.111' .1111 .1111IllpilL)

isluatuSuit atp Jo liv JO our iii.b:0.111 alp S ul puuJ(I itzut utp JI/1 4I: sandor) 1uJUl.i U.i 1:41:1 alp 111:,).)(1 's.D;

bvc Swiuur i.1111 111.t.)4 :stitia.tIlli Jo Sup .1111 tit) szo,)tioliittio.) 10.0 )4

asotim Xltuuuu Utt luau; 'Autupuual oti1 `111)tug1:d 1st ,u11 paimaascp o.%%

fivatuow alp .103 'Jr -luour.v,I .qadlauub timouluu 11011niv *.plotuX14.1 Spolartib :Om puu 'Spolainit) poputtuduwa 040.1 aq1

in 14! ls...)1alut 31 .1r:010411c um: ivtipupti 01 RI Z JO ill: anarti.)19i) Spi101dttIOD 01 sei.ruJX 5 .103 sti111001 tpla JO 1-fiquup,)(1 11; s111)11ISvd rubo

wpm' aa.W0 pull '45113 ()wo 111.10.1 c.itt!pipiq .),uciaanti

*Slavutn

r)tL 10 ÒILO. = 31. %;,-! 11111i111011 011 '811 .10,.1 111

LL :!:)1J,VIVAILLVIN:

78 THE SixTH 'YEARBOOK

ohliaations, he %ill pay $500 at the end of each 3 months as long as necessary,and will pay whatever final installment is necessary to close the trans-action 3 months after the last $500 payment which is required. Find (a)how many $500 payments will be required; (1.0 the final smaller payment.

Noiatin. (a) If it payments of S500 each ,re exactly sufficient to dis-charize the debt, then the present value of an annuity of $500 paid quarterlyfor n periods would equal $10,(X)0: that is, by equation with i.=.02,

10 000 = a' .02).(a, -I at IV) (ti)

Front Table IV. we see that I at .02) 19.5, which is lessthan 20. lIenee. 25 payments of $i500 each would not be sallieirntto discharge the debt. Also, from Table IV, (a:,.7, at .02) 20.12;hence, since this is flaw, than 20, it follows that 26 payments of$.500 each would be MM.(' than is required to discharge the debt.These facts show that the debtor should pay 25 installments of$:100 eaell, anti some amount W, Icsx than $500, to close the trans-action at the end of the 26th period. The last $500 is due at theend of 25 periods. or 61.,'t years.

(h) To determine the unknown final payment, W, we proceed as follows.First. we solve equation 6 for a by interpolation in Table IV. We find that

4.763n = 2.5 + = 23.7975-a,moThen, Theorem I. which follows, states that the final payment is 500(./975),or $398.75.

The solution of the last example serves to illustrate the fol-ImvinEr, theorem.

TittIttta:m I. When a ciPbt. A diseharzed, principal anti interest included,by payments of R at the end of each interest period for as Ione its necessary,with in smaller payment one period after the last installment R,then the [lat., :1111 size of the final sioall payment can be found as follows:

1. Sol', R(an." at i) fair n 1771 ;at( rpo/ation t» Table IV.2 If the fil)tined it n k f, usher, k is a paiitire integer and

pa11;,-,- thaa 1. Ihrn the finql pomrnt fiqe at the rndor ( k 4- it >iife-...4 1a rio,IA and

final porn, at f (7)

The essential part of the statement of Theorem I is ill equation7. whose proof is beyond the scope of the present chapter."rquation 7 w:Is used in the solution of part it of Evample 6. In

"Fnr a r.1.1% W. I.. Hart. Amer...7h iffottly, Vol. XXXI," I.1:#291, p. 373.

-iXt).1.11.1; At. 1F.6.1 001:1 '! .1 '01 '1"

-10 1) "1

..; .1\.. it

0n0.01 it

000.01

:11 \w(I)1,11; it s"1.: 11A 4. 11-!11'111'.' 0-11

LU ..:Hunutn;!wos poputimitlio.) ;,1 .11poutio.o; 1.1441.4 ...411:!:.

Jo 1111.1 0111 11: 1q1) io .1441 ..1)!.v..1,1 1.1110 ..111 -)-10.. -('( (KX)*()IS Ot.1 Istitd 11.11tuttli: 0,011.\\ (.111: 7.07 111.1.1 1.)11 1 01.1.

eitt4utiop tinj 1.1111.T1!..: otii t!! ipitimi ,11 101

.00:z." .1" 0(K). 01 ) CO.

i1 'a( .11141.10 .,III Di 01( -! 11-1pIAA 9 II' 4.: I l't tilkirp11,1:4 ,10:;?'1(,S

4( i .1' "11 4(4 Or it 3.).1l: 11 .1(1 it 1,1, 11

I() 0411111Xa (11111111:911.! 01 .1111 4.. 014;11 41:

4ir:0.111 1,141; .7111:.)110.: 0111 .1(.11 p 7.17. JO 1,11

Jill II: 411,1111111:011! 0110 til 0111 1)4 141114 JO ..41: 11 iv .111:1t(1111.

4:.:J.1J1u! ..iud tit tioo.t7114 p1:4! (xx).01;.:: 114:111 v

'Ily.t1J1).110.11 fu 111.10) J() pud no fi(I 17*

LI) dluintun.),n) ()) rldpi.11 eqtr,yare .LOf psodo(f .14 tio Jlrl, I..)11)411 ,)(11 j0 duitlidad L)1) 1)) 111.m4iind 10)

:sm,)41:601.,01104 0114

30 ulus 0114 s1 4(101) 3111 10 411140.4,411 110 0411)(Ixo J!po!.10(1 5.10411311 0111

..tonpo.t.) spi 04 4....zat1111 .P.[ 4 1111 1 ,s.irp mulls mil 110 1)0111 :3111

i111 1.11 :.:11:z0t101) 11::11/0 14.1[1:111 J04E1011 0;1 1 11 : : {1 ,H1111-r-0 HUT:, 0.11

.pult 4atu s1.114 101)44.1 .poqi.)/a pun/ em.r.vi.ix 011 411o1'

-1411, 04 1411 :s .t 011 '1)1111.1 .10.11 111114 1: j0 1109110.1.1 011 1 111.1111.1

0.111 1111 0114 .103 ::1)1A(4.1(1 .10411)1) 0114 11 *::.11:0. jo IWO 011 41: 40.1111

Jul) 111 Null Jti ,19 1 orli pul: '0141) :.11 1:.i4'ti

rl 44at.)4tit 4 101 I 0)tu.).1.o11: to .1).).t0.1.10(1 ri; 11441 ...:(41(111.:

.11,:0(1.11)

0114 p3111.101 ,1111 i14nn1111 -1 .411111 .1114 10 1)11111

)l(4

-01; .c1:11

111

,S1

01

411110100 .)111 'r.:111. (1.11) .1111111.1.111 111111) :r.111 1S.).1111 11.111:11111401

1/11111 1: 11 '01 1:1) ,).1111111 01110:4 10 01111 .-31111[1:1 1101 11:1;11110 111:

1),)}1;11111111,).11: 141011 1-1 1111111 .7111p1111,": V"spung 21.1111u!s

(101414111,0 01

1),)101+ tit 1)11 11 'f ipim .ioodold dip ,).1:11 pm 1)111011 if

1011111110 0114 '111(c1 '1.101 11:10(1.10 1111 .11 1)01111:1qt) 14.1k 11.1111.11 110111;11).)

011 r:0 01111.:S 011 4 .ii,11:311 11 11:(111.))11 0.0 1)11111 t\ 14p140.10.1

4!JtitIxo riin.01(1111,) `poq4,-,01 oputut.n:4-101 .:"11,11 1- itot01(11)0

0114 30 tio!litios ...zsoautti 110!41.110(1,tolln 1141 timi0.111 p.TI!upto '4(41 d.1

131:x0 nu .10 1101 n4.1 1:4111p 'iti!ls,),L)111! .1.011()1141 ,).vii 11090111).1

62, .1,1IVjS"..1AKI :10


Bonds. A bond is a written contract to pay a definite redemp-tion price $1` on a specified redemption date, and to pay equaldi? idend$ SD periodically until after the redemption date. Theprincipal SF mentioned in the face of the bond is called the facevalue, or the par value. A bond is said to be redeemed at par ifC = F tas is usually the ease, and at a premium if C is greaterthan F. Ti e interest rate named in a bond is called the dividendrate, or the bond rate. The dividend D is described in a bond bysaying that it is the interest, semiannual or otherwise, on the parvalue 1: at the dividend rate. The following paragraph illustratesthe essential paragraph in a bond:

The Kan,as Improvonwnt Corporation acknowledges itself to owe and,for value reveivNi. promi:es to pay the bearer Five Hundred Dollars onJanuary 1. 1936. with intre-t on the said sum from and after January 1, 1925,at the rate Cr 7c per annum, payable semiannually, until the said principalsum is paid. Furthermore. an additional 101:*;- of the said principal shall hepaid to.the bearer on the date of redemption.

For the bond of the preceding paragraph, F = $500, r = $550,and the semiannual dividend D = $15, which is semiannual interestat on $500. A bond is named after its face F and dividendrate. so that the preeedina; paragraph is an extract from a $500.Gr'c bond. orrespnmling to each dividend. tlu would usually beattached to t:.0 bond an individual written contract to pay SD onthe proper date.

When an investor purchases a bond. tie interest rate vhich hisinvestment yields is computed nuder the assumption that he willhold the bond until it is redeemed.

WI:en a bond is sold on a dividend date, the seller takes thedividend which is due. The purchasi'r receives the future divi-dends. which form an annuity whose first Dayment is doe at theend of one dividend interval, and vhose la-u payment is Inc onthe redon,ption date. It' an investor desires a specified yield onpurchasing tl:e bond, the price $P which he should be topay is by the following (quation. where prezwnt values arecomputed at the inveAinent rate which t!:e I'llyer has specified:

Iprc,ent abi of th,- ltion 'tate)Ht r,11,11. tbe annuity formd by tbe diridend.I. (8)

Ex Amptx A. A $10(1. el' bond. \volt piyable semiannually, willredee'me'd at ir at the end of 15 years. Find the price of the bond at


which a purchaser will obtain a return of %, compounded semiannually, onhis investment.

soto;oa. The send Imolai dividend is !i3. The it MrtiOn is $100,due at tie. end of 15 years. The term of ihe divider I annuity is 15 years,or 30 interest periods. By use of ectuation S,

1' 100(1.02) 3(ii.;.lat .02) $122.40. (9)

Iu tlitiatiull 9, We usvd 'nail', 11 ;did also Table 1V.

Determination of Bond Yields. A purchaser of a bond can -nut, (IS a rule, d' 'tae the !wive which his will pay, er the interestrate whielt the bond 511 :111 yield, although, of course, he may with-hold his purchase until the ondition,: satisfy him. Essentially, apurchaser of a bond must pay It !ie jS specified by the lawof supply and demand. :t' exemplified in the prices on the bondmarket. Hence, We are led to the problem of determining the in-vestment yield is given by the pureke,!. of a bond at it 1.2..iven

price. This problem is the coot 'erse of that treated in the precedingexample, where a yield as given ;oat we determined the price..1 :solution of this converse problem is too lengthy for the presentdiscussion, although the solution presents no serious diflirtilty. Ifall annuity table, like l'able IV, is available, the problem can be

by interpolation, alter it suitable arrangement of prelim-inary details. "che solution is particularly easy if a bond tableis available giving the prices of Inanis at various yields and forvaruts times to the rellelliptit)11 Batt'.

An approximate solution of th problem of detertninin!, a bondyield can be obtained by the method ui the next two examples.Example 9 illtistratts,.: tilt' of a bond bought at it discount,that is, bought- fur less than the redemption payment. Example 10illustrates the case of a bond bought at a premium, that is. boughtfor more titan the redemption payment. It is interesting to notethat the Met 110(1 ": of IlieSe eX:11111) les 111VOIVe74 mere arithmetic.

f:x \mils. ¶1. A $1,0u(). 111,11,1 pays lividripis ;num:illy and is redveni-101111: at par at tin. cod of 10 y, If tke hood ho11:2.1it fur :;1550, determinean al piovinat ion to the yield winch the in\ ,..tor obtains.

do aknil it ;4)0. iit in% t -1,1- pays for die bond.and recii\ at the end of It) -ear,, beside: the nds. The:

(Ledo $1.;41. ieresents die leelinailated .iiii. :it the end(1! It) y.-trs, ''I I all t Iii % thAt %%11;e1t the

po% 1)tirtm; tin' 10 yt.:11,.. flip itiNTstur Atii11,1 think of tile value ofrt jostiliont ion of to, liwthod. wthn of

in, lit. 1'.

S2 THE

1)1,2 bond as inere.isait: from $ti:51.1 so $1,000. 11 the $150 inerease yore spread" r lit.- 1() vtir-t, filo inert r ir 11,1(150), or$15. This $15, r lii it' animal of' 1410, iii $7.1 in all, isappro\itnatt 1 .00to ti oc. 1.1 "I 11-1111 ti lruitn I ho , 1.1, "iinge\Ay iii thi. 120,...,0-Flom,

1)1tip' luau hu'i-u-.tu 10,, ::,,A(1 lii 711.0011. 111. ittIt-rt-t-t rat,' obt 611111 by theintestiit is ai proxstn.ilidy nit,- at w,litolt $1'5 is thi( mteryst for eine yvar

. 75011 $1125; tin. ran( I-925.

or Thai i-, thu bona Yl(.41,4 approximatelypot* yoar.

It ran In' ....lit) \VII 1 Lai I he method (II i'-1'-: example, fur-nishes :t \vilis.ls is err(ir not titan .2'; . etSa very extrenic !halt, in the eustoniary type of problem.

xtNspi.t-; 10. If ilii 1)((n1l of Ex:Law'', 9 hula:lit for S1.200, find anapproximation to the i'(1,1 °lit:lined by till, inv.tor.,"i'llie)te Earle diva!, nu! k $400. '111( pay- $1.2(XT, reeiivisidy t(s1.t()0 at 'ow ,.11,1 (if 11) diviiictids. IT 14 nut earn el

to say that the iliffi.ri(nee 0.2u0 1,00(P. rfprt.s(-111:, pritt-(11,31. w,. liuuttltt s.t tit tt this $2(X) ri pri tits a r..t wit of prifleipaltlr.t.lt 11 1-: tltro:tult titt tit% It1 payttit Th. rofor,. t-tt.lt divitIontlonly partly of ; part ui t-ttli uIi tibtuul 1.?. .t rtturti prineipiel.If tilt $....100 siisirti of ;Thr. -:jit inilforto1v14 over dn.( 1t.) yrat'-. therilurn \'-:u1 1),- ((r $20. Ii' nee. in of :160,tip Is a rt.t11111 ui abolit $20 (4 prineipal; iii.' $.10, IS inti-rest.

The v.111:, (,f from $1.20) to $1.000 di, 10 years;the averatto \ able is 1211.20e) -I-- l,()0), 'it' $1,104), Tip' inti-rost rate whieh theinvistor ithtams is approxiterit, lv iii r:itt. at which $10 is interest

.10fin. one yoar on $1.114); 11114 nit, i .04. The Inv( stir obtains1,100'apprt)ximateli 3.0".'i ht' invpstnictit.

r V. it(lisetIssiim of Part, II, III, anti IV wa, aroinitt an

cxtrrniely slim!! mahl.t.i. of Holation,:. !orIlacc for tvo for coll.polind a:1.1 1%.;) foratuwitirs. Iii racli ronsinireil, t1.1( sii:10 ion started, iiireetfront one (if these fundamental eq;Piti((ti. i- called titdie relative situp:ieity t. stn.lt :t flied:1)1i. ill viintrit4t to one %vliere\ve \vennit entp:oy the auxiliary 5i.1 lit solv-intl., in tut'll, ill cavil equation. for cacti syinhen :11 terms i)f theialter:4 present. inct1.0,1. :ipplitci to violation

.\ 11;:(;1:. ((- t1- 1,-t -1.r--1-1 .!t-'.:. tr rho E.:0 --I' I.. 1tni1." t-. .ti ;)3i tr I u. ma,.

NIITHENI.VIsICS OF IX VEsTMENT 83

r)k, would demand that we solve the equation for r, andfor k, as a ill.rliminary to our cli,(.11:-.-ion of comp)tviil interest: the

%vould

log S log P I,S= r 1log (1 + 7.) '

'file addition of equations 1 to the two simple formulas which weinivo used in diseussingu compound interest Nould give a ratherromplicate11 mathematical bakground to an otherwise simple sub-jet. In nuts II. III. and IV, the restriction of our formulasto the few simple equations whirls WC employed gives :t mathe-Inatical hark.ground which is easy to remember and convenient to

The methods employed in this eliapter were absolutely depend-ent on the use of previously prepared tables of the values of thefundamental expressions involved. Failure to u,(. such tables tothe fullest extent NVollid reStlit ill ;111

Contarts with actual practice in th.)se fields where the sub:wetfinds Ilse.

The author lwlieves that a large amount of the subject matterof this chapter Il1erit!4 (.011Sideatit)11 OS a part of any advatiredcourse in arithmetic given ill the senior high school,


TABLE I(.1 i) k

k J

1

23456789

10

11

1213

14

1516

17

1819

20212223

242326

272829

30

-31--3233

34:35

:36

:37

38:39

40

11:3(7f 2%

1.0150 1.02001.0302 1.04041.0457 1.06121.0614 1.08241.0773 1.10411.09:34 1.12621.1098 1.14871.1203 1.17171.1131 1.1951

1.1667F-- 1.21901.1779 1.24341.1956 1.26821.21:16 1.293612315 1.31v31.2302 1.34591.2(390 1.37281.2880 1.40021.3073 1.42821.3270 1.456S1.3469 1.4859.1.367 L- 1.51371.3876 1.54601.4081 1.57691:1295 1.60841.4509 1.61061.4727 1.67341494S 1.70691.5172 1.74101.5400 1.775S

1.56'31 1.8114--1.;WT 1.8176

1.6103 1.881.11.6315 1 .9222

1.6590 1.96071.65:39 1.99991.7091 2.03991.7:318 2.08)71.7608 2.122:31.7872 2.16171 .S 140 2.2080

TABLE IIll i)-k

1 .985 222 .970663 .956 32

456789

.942 18.928 26.914 54

.901 03

.6771

.S7159-10- -.W6167

1213

141516

171819

.836 39

.824 03

.811 S3

.799 S3

.788 U3

.776 39.764 91.753 61

20 .7i2 1721 .7315022 .7206923 .710 04

21 .690 5125 .69 2126 .679 02

JIGS 992S .659 1029 .619 36

3031 .6.,0 3132 .t3:20 t);)

33 .611 8234 .002 7735 .59:3 S736 I :65 l)373,S

39

40

2%

.980 39

.961 17

.942 32

.923 85

.905 73

.S87 97

.E 70 56.Ci3 49.836 76

£20 35-. . -

.S01 26

.788 49.773 03

.757 SS

.743 01

.72S 45

.714 16

.700 16

.686 43

.672 97

.659 78.616 84.634 16

.621 72

.609 53

.597 5S

.58.5 S0

.37.1 37

.36311.6.39 , .a52 07

.541 2a

.5:30 63.520 2:3

.510 03

.500 03

.490 22.576 11 .480 61.567 9.2 .471 19.559 5:3 .461 93

-.551 .452 89

(11, INVEsTMENT 85

TABLE III TABLE IVs;71. awl

It 2','.

1 1 tom 1.00002 2.015(1 2.0200;; 11.0452 3.0604

4.0999 4.12165 I 5.152;3 5.20406 6.2296 6.3081

7 7.3230 7.4343- 132.-3 8.58309.559.3 9.7546

10 10.7927 10.9-197

11 11 86.33 1 12.168712 I 13.0112 13.11211:; 1 1.2368 14.6803

11i

15.1501 15.973915 16.6.21 17.298416 17.9321 18.0393

17 19.201 1 20.012118 21.412319 . 21.7967 2*, 106

0 24.2974

21 21.170) 20.7~3325 8376 27.2990

0 27.2251 25.8450

21 28.6;3:35 30.421905 30.06:10 32.030326 31.5110 :33.6709

o067 35.31132" 3 1.1815 37.0.512

:35.9987 58.7922

.30 .37 40.5681

it 39.101 42.37940. I 4(1 ; 1.2270

42.296 6.1116:;1 4:1 93 31 45.0:338

15.1!+'21 19.9915:36 17.2700 ;31.9941

9..51 5 1_03 1350.7H,11 56 1119

:39 52 107 58 2372

40 5 1 :26711 60.4020-

;

1

2 1

3

45 I

6789__

10 ._11

i

1213

141516

17is19

20212223

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MATIIEMATICS IN AGRICULTURE *

By HARRY BUR ;ESSfnipt rxit I/ of Minnesota, Dupartment of Ayrieutture,

St. Paul, Minnesota

IN

Value of Mathematics in Agriculture. There are still manypeople who question the value or the need of any appreoiableamount of math( au,tiettl training in the field of modern agriculture.On this account. at the outset, it will be well to consider the presentstrueture and relationship.) of the agricultural industry.

The Business of Farming. The modern farmer, if he is to besuccessful, must be a business man 'n the fullest sense. Ile musthave in his mental equipment a I argi. and broader knowledge ofscientific, economic, and financial principles than that required inalmost any other hue of business in the world to-day in order thathe may plan, direct. and carry out the operation of his farm intel-ligently and with profit. Ile must be able to plan the size andarrangement of buildings on the farmstead, to subdivide his farminto fields of the proper shape. size. and arrangement for economi-cal and balanced operation, to plan suitable rotations, to inaugurateanti carry out systems of fertilization and pest control, to builda farm calendar. anti to schedule the amount and distribution of allman and horse labor. and mechanical power. He must know some-thing of the relative efficiency of inethods and equipment as wellas. of the principles of depreciation and replaccInclit of bothmachinery and livestock, lie inu4 balance his crop productionto meet the needs of his livestock. and he must plan his feedingrations and seb.rt atn f breed new stfains in both crops and livestockto insure best resirts in production. Ile must be able to estimatethe amount of paii.1 required for a buildin. or the best shape orsize for a given geld to be fenced with a known amount of fencing.lie must know tit- capacities of men aaIltl 111:101.111Vry , of bins and

%1 [qu!' tor the Yral 1),0,4 Pit Nqt,,,n,t1 counell of T <'acArr, of Math-matirx and IIPPr"vod h) Or Dean and Nrrlr as M bwellaneou5 Paper No. 2:26of the 1 !IR ersit y of Minnesota. Depart ttont Agriulture, Attgtist,

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least potentially ami fundamentally, essential to training in agri-culture and agricultural science.

Especially closely associated with the field of agriculturalscience arc the various milling industries producing flour, food-stuffs, and clotliing inaterials. as well as paper, lumber, and otherbuilding and structural materials. This combination creates an everincreasing deniami for Ultimate knowledge of modern forest de-velopment, conservation, and management, of salesmanship, ofmechanical knowledge and training, and most important of all, of

including such matters as investments, insurance, bank-ing. property valuation, luaus, and amortization. The range ofnuttheinatic..1 knowledge required to handle this varied demand isreadily seen to be all the way from the four fundamentals ofcommon arithmetic up through the various ramifications of thecompound interest principle, mensuration, logarithms, statistics,calculus, and the mechanics of materials.

The Rural Home Problem, Inseparable from the conduct ofthe farm business s the factor of the rural home which involvesproblems of housing anti feeding adjustment as between the op-erator's: family and the hired help, the simplification of laborthrough modern methods anti devices, sanitation. the economies offood and clothing, and child training and NN cl fare. Here the re-quirements in inatheinatical knowledge and raining seem not tobe so broad and varied, but they are none the less definite in thefields of proportion. mensuration. anti minor financing.

Agricultural Science and Organized Research. This fieldcomprise:::

A!;riculturul Biochunistry. which to-tiny is at the very founda-tion of human food !old clothing supply in the pocesli:mg ofdaily products, flour, and other grain foods, dye stuffs, and cloth-in:, materials: aunt also 1)11.11(1111LT, 1:1Ztt IS tit`VC!OpCi 1 11'0111 farmgrox%11 crops. and the

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Ti:. B:clouical which involve the development of de-sirable types and str:tins iii plant anti animal life, such, forexample. as are resistant to diseases and climat ic rigorsandthe control of both plant and animal pests.

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sought; while the selection of the type of tractor best suited to dothe work on a given type of farm under given local conditions re-quires a knowledge of structural materials, depreciation, soil dyna-mics, and labor and power distribution, which is unique in thefield of engineering,

It is here interesting and significant to renumber that Dr. I:, A.White,' the first acknowledged agricultural engineer to receive aDoctor's degree in that field, developed some fifteen years ago ashis Doctor's thesis "A Study of the Plow Bottom and Its Actionupon the Furrow Slice," a scholarly treatise of 34 pages, 24 ofwhich are pure mathematics of a high and difficult order, Thisstudy has lend a profound influence on modern plow design andusage, It is the recognized forerunner of extensive studies, nowbeing enrried on at sonic of our agricultural experiment stations, onsoil dynamics in relation to tillage as a guide to the efficient designof farm power units and tillage machines required in modernagriculture.

The intricacies of design called for in modern scientific plan-ning of farm drainage and irrigation systems and of soil erosioncontrol works, involving not only an intimate knowledge of thebasic laws of water flow in closed conduits and open ditches butalso of moisture relations of crops, of soil texture and soil watermovementsoil hydraulics. if you please call for a grasp ofadvanced phases of mathematics and mathematical physics in newand difficult paths that is even now taxing to the utmost the abili-ties of our leaders in mathematical and physical thought.

EDUCATION IN MATIIENIATICA FOR AORICULTIIRE

This brief survey of the field of agriculture in its various rami-fications brings us at once to grips with the question of whatfundamental mathematics in secondary school education is nettledto prepare for work in this tieltl. In this connection reference mayquite properly Lie made to a former paper on this subject, rend be-fore the Mathematical Section of the Minnesota Educational Asso-ciation in 1021.3 The situation is not radically dif,..erent from whatit was at that time except that perhaps the demand for higher

t 14rpvtor of top National VonnultteP on Thp Relation of Meetrleity to Agri-culture.I hi"rPisi aJ Agrirnii low/ ran 1,, No. . 1'0l. XII, pp. 111.182." M1oltoom MathputatIval ItoquIroment for Agrlvolforal Study."

o,utlen Trauhr. , Vol. XV, No, 1, January 1821.

MATHEMATICS IN AGRICULTURE

mathematics in education for agriculture and related activities hasgrown much stronger.

Historical Retrospect. During the first decade of the presentcentury this demand was nt n very low ebb, the chief value ofmathematical training for students in the agricultural field beingconsidered by ninny leaders in that field as a necessary evil whosechief value was cultural and disciplinary. This attitude, however,resulted in such a decline in the general quality of scholarship inagriculture that by the middle of the second decade of the centurya decided protest against neglect of mathematical requirementbegan to make itself felt and its inure vigorous proponents started adefinite study of, and a propa gamin for its more utilitarian aspects.By 1918 this activity began to take definite form in the claim thatwhat was really needed was a general inatlkmaties especiallyadapted to the elementary requirements of agriculture. All non-essentials were to be swept away and only those topics were to beretained which could be shown to have a direct bearing upon sonicphase of agricultural work, supported, of course, by the recognisedprimary fmulamentals.

Within three or four years several new texts appeared purport-ing to meet the requirements just stated. The general plan seemedto he clothed with official sanction when the Resident TeachingSection of the Association of Land Grant Colleges and Universities,in its annual convention of 1920, voted approval of a syllabus fora suitable text for such courses, which was presented to the session,in galley proof form, by Professor Samuel E. Rasor of the Depart-ment of Mathematics of Ohio State University.

A Specialized Mathematics for Agriculture. There seemsto have been quite general agreement as to the requisite detailsof such a course, these being in general about as follows:

A review of the imzie essentials of elementary algebra as givenin standard secondary schools through quadratic equations.The elementary principles and practice of common logarithms.Elementary series. including especially the progressions and thebinomial formula.Depreciation and elementary accounting.The compound interest principle and annuities, perhaps betterthought of as the mathematics of investment.Ratio, proportion and variation, dairy arithmetic, and mixtures.

92 THE SIXTH yrAnnoox

The principles and practice of numerical trigonometry, espe-cially as applied to general mensuration and plane surveying.Permutations and combinations,

Elementary statistics, frequency distribution, and the elemen-tary theory of errors,

Consideration has also been given to various combinations ofcertain of these details, under headings of certain fields in agri-culture, such as, for example, the mathetnaties of farm manage-ment. The inclusion of this type of material has received eon-sidernble support from the specialists in the farm management field.

There has been a tendency on the part of some of the earliertext writers to include topics that belong specifically to other fields,such as drawing, physics, and surveying. This practice is not welladvised, as each of these is a field by itself that holds and requiresfully as prominent and important a place in our school and collegecurricula as does mathematics, While these fields are among therichest sources of real problems in matheiaties, the inclusion in amathematics text of chapters on pure physics, drawing, or survey-ing is to lie deprecated, as no one of these distinct fields of study oreven of their major subdivisions can be adequately covered in atchapter or two. This seems especially trite when it is rememberedthat the time allotted to mathematics in school practice is fre-quently too limited for thoroughgoing results even in that fieldalone.

Difficulties Encountered, The idea of a specialized mathe-matics of agriculture has grown up with such startling suddennessthat, as might he expected, mistakes in the plan of operation haveseriously inteijered with its complete success in practice. Two ofthese mistakes stand out with unusual clearness.

1. College administrators, continually harassed by the demandsof numerous new types of courses for admission to the requiredcurriculum, and often influenced by those who cherish a lifelongand unreasoning antipathy toward mathematics, have seized uponthe new idea of mathematics in agriculture as an excuse to limittl:e time allotted to mathematics in agricultural curricula to adegree utterly inadequate to the around dint must be covered. Inmany cases this has been carried to the extreme of limiting thetime allot. mathematics in the agrieultural college to onequarter, with three to five hours per week. when those responsible

MATHEMATICS IS ACIHICLI.TCHE 93

for the conduct of the work are painfully aware that three to fiveclass hours a week for a full year is little enough time in whichto expect the general run of freshman or even sophomore studentsto grasp n working knowledge of the mass of topics called for insuch a enlirAC. No doubt the administration justifies its attitudeIty the plea that the useless lumber of old-fashioned mathematics,eomprising fully two-thirds of the subject matter material, hasbeen cut awny, and that I. should be possible to cut the allottedtime correspondingly; a not well digested argument to be sure, butintrenehed behind a strong rampart of power.

2, Too often the teacher assigned to the course is selected be-muse of his training in the field of pure mathematics, and fromthe general faculty of the department of mathematics. In suchcases the chances are that, although he may be well trained inmathematics and even though he may be an excellent instructor,he knows little of the field of agriculture and his vision o. the rela-tion of mathematics to the agricultural field and its ninny-sidedfunctions may be represented by the minus sign, Regardless of hisscholarly attainments. his pedagogical talent, or his enthusiasmfor pure mathematics as such, a teacher of this type will find itvery diffieult to elicit the interest and support of the various de-partments and groups of an agricultural faculty so essential tosuccess in teaching the mathematics of agriculture, and he is thusVPI'S apt to be deprived of his most reliable source of real problemswhich show the relationship of the pure fundamental science to the

Selection t Teachers. It is a well-established practice inengineering schools and colleges, based on experience, to select asteachers of mathematics men who have been trained as engineers,and are, as far as possible, experienced engineers with a naturalgrasp of the relations of put e mathematics to the problems ofengineering. In the light of that experience and with the fullervist in of the place of mathematics in agriculture that. is openingup to the present age, why is it. not the sane and logical procedureto select our teachers of mathenutties for agriculture more largelyfrom the ranks of those agriculturists, agricultural scientists, andagricultural engineers who have a clear understanding both of pureinathematic's as such and of its relation to the problems of agri-vulture? This would seem to be a rational practice, not only forour colleges but a; ) for those secondary schools where preparation


for advanced study in agriculture and in allied lines is a pro-nounced feature,

In fact, in view of the time limitations on mathematics in thejunior agricultural college curriculum, it would seem that the ulti-mate success of any type of instruction or any typo of course inmathematics of agriculture must rest largely with the teacher ofmathematics in the secondary school,

Sul*.ct Matter Requirements. This thought then bringseven more definitely before us the fundamental question; Whatare the secom.ary school requirements in mathematical study thatserve as a suitable preparation for advanced education in the agri-cultural field? An additional decade of experience in this fieldhas not greatly changed the answer from the one given ten yearsago.' The extent of mathematical training needed is largely de-pendent upon the particular student's line of specialisation in thefield of agricultural or related study, but for good results thesecondary school mathematics should, without doubt, be much thesame in all cases. It does not appear that its outer boundariesshould greatly differ from those usually existing, in theory, in ourestablished grade and high school curricula, However, possibilitiesof improvement over the stereotyped old order in interior sub-division, topic arrangement, content, and methods of presentationunquestionably exist.

Arithmetic, The young people of the present generation arelikely to he bunglers in their work with common arithmetic. Toomuch stress, therefore, cannot be laid on extensive practice in theearlier grades in the use of the four fundamental operations; andthe numerous short cuts in multiplication and division should beknown and used with facility by every normal young American.It is wholly unnecessary to find a supposedly normal child of highschool age and standing using the process of long division in divid-ing by any single digit from 2 to 9, and no teacher should acceptor lightly pass over such a piece of work.

Teachers in other fields of work than mathematics, in whichmathematics is a necessary tool, complain that the students areincompetent to do work involving the common operations in frac-tions, decimals, percentage, and simple mensuration. The responsi-bility for this situation cannot be laid upon the children themselves;it, rests upon the teacher.

Mathethat lea Teacher, Vol, XV, Nu. I, 4oututrY 19::1.

MATHEMATICS IN AGRICULTURE 95

Considerable stress is laid by leaders to-day on intuitive orinformal geometry as though it were a new idea in mathematicaltraining and usage. It is not new except perhaps in name, Thearithmetic texts of forty or fifty years ago included a very con-siderable chapter on mensuration, There are still living thousandsof our older people who received the fundamentals of their educa-tion in the one-room, ungraded, country school of an earlier day,and who found both intellectual profit and keen delight in themastery and prarticoi of both planimetric and volumetric mensura-tion, long before they ever heard of geometry as such. Many ofthem who never heard of formal geometry, as now taught in our highschools, can apply their early training in mensuration to the every -clay problems of capacity, area, and direction with greater facilitythan can the majority of their children and grandchildren whopassed through exposure to arithmetic without any conscious con-tact with mensuration and who consider the mastery of geometricalprinciples to be the epitome of drudgery and dullness instead of theacquisition of new mental power which it is.

In brief, then, the teacher of arithmetic should insist on a readyand correct use of the fundamentals in practical applications.Intuitive or informal geometry is as properly a part of arithmeticaltraining as is percentage or interest, and beyond that it is a fruitfulsource of practical problems to which no student in the agriculturalfield should he a stranger.

Algebra. As elementary algebra is only generalized arithmeticthe beginnings of algebraic notation and processes, such as theterm, the exponent., the simple algebraic fraction, the binomial,and even the simpler special products and the simple equation inone unknown, should be introduced to the student of arithmetic atthe earliest possible moment. There is no reason whatever whythis practice should not begin at least as early as in the seventhgrade and possibly in the sixth. Then by the time the pupil reachesformal algebra in the eighth or ninth grade he will be ready forit and the Jansition from arithmetical methods of thinking innumber will be gradual and natural and carry with it no depressingmental shock. This is as it should he and is in line with progressivethought in education. Dr. Reeve's quotation from Dr. Charles W.Eliot 5 sums up and clinches this whole matter in excellent fashion.

Fourth Yearbook, National Connell of Teachers of Mathematics. D. 141.Wirt% u of Publications. Tettchera College, Columbia University.

oe THE SIXTH YEARBOOK

In regard to requirements in elementary algebra, this point hasbeen so definitely covered in the former paper already twice re-ferred to that it needs not to be discussed here, although someadditional comment may be justified. It now seems clear that themore complicated factorable form a" ± b" should be among thoseomitted from high school work in algebra for it finds very fewapplications in practical applied mathematics. The time thatwould be spent on it by the high school pupil can be spent muchmore profitably on other parts of the subject..

One bit of repetition from the earlier discussion seems amplyjustified, even though it is in direct opposition to eminent modernauthority. Treating a proportion as ar equation of simple frac-tions is, of course, perfectly proper, fi.r that is what it is, but itmust also be treated and taught OP ,t proportion, The principlesof proportion are essential, simp'e, and eminently useful. Theproportion, as such, is much more readily applied than is thealgebraic equation of simple fractions. Proportion is the frontdoor to variation and both are essential in chemistry, in physics,and in engineering. The `cocker of mathematics will not secureand retain the whole-hearted cooperation of the scientir+. and theengineer by refusing to teach the principles of elementary mathe-matics for which these groups have daily and constant use. Thechemistry of to-day is hard for the average beginner anyway.Whether the administration is willing to admit it or not, beginningchemistry is an elimination course in the freshman year in manyof our colleges. The teacher will make it doubly hard for theaverage freshman and almost insure his elimination, or, at least, hisstrict probation, by refusing to equip him with a comprehentiVeand thorough knowledge of proportion and variation; and tnisrefusal will not magnify the teacher in the esteem of the student.

Elementary series, particularly the progressions and the bi-nomial theorem, are the very foundations of much biological scienceas related to agriculture. Therefore, if they are not tall:tilt, andthoroughly, in the secondary tchool they will have to be taughtin the junior college. Is such a plan good pedagogy? Is it notrather a waste of valuable time for the student to have to secure,in the junior college, essential mental equipment that he shouldalready have when he first enters there?

One feels an urge to enlarge on lie place of the graph in algebra,especially when he :onsiders its multiform uses in agriculture, but,

MATIIENIATICS IN AURICULTURE 97

as far as the subje.t is related to the teaching of mathematics,Dr Reeve effectively sums up the essential idea as follows:

The study of the graph is a major trend to-day in algebra because, withthe formula, it helps to clarify the idea of functionality. We now emphasisethe meaning of graphs rather than the making of them. . Owing to theprominence of the statistical graph, and the increased interest in educationalstatistics, graphic work is assured a permanent place in our courses in mathe-matics.'

It might be added, "particularly in those on the mathematics ofagriculture."

As algebra is basic to all other mathematics it may be advis-able to emphasize the fact that the high school student should beso well trained in it that he handles its elementary principles withunderstanding and facility. If the customary year allotted toelementary algebra in high school curricula is not enough toprohce such a result, an extra half year of advanced elementaryalgebra, treating of the more difficult forms and including addi-tional topics such as the progressions, variations, and logarithmsis desirable and preferable to so-called high school higher algebrawith its abstract proofs mostly beyond the intellectual grasp of thesecondary school sttalent. I' a. the young student facility in appli-cation conies from I andling often, with familiarity, rather than frontdeep reasoning. A igebraic demonstration largely calls for greatermental maturity than is usually found in the high school student.The practice of some of the best high schools in offering the sug-gested extra half year of advanced elementary algebra or generalmathematics based on algebra, in lieu of the half year of so-calledhigher algebra, has been productive of good results and is to becommended.

Geometry. Elementary fundamental requirements in geom-etry, as in arithmetic, algebra, and possibly trigonometry, are muchthe same whether the ultimate field of the student's life work bethat of medicine, law, engineering, theology, or agriculture; foras one graduate school dean was recently heard to state, advancedresearch in applied science can be naturally and safely based onlyon work in the fundamental sciences of mathematics, physics, andchemistry, with particular reference to their application in solvingthe given special problem in the given field of applied science.

The fundamental requirements in geometry in secondary schoolblurt!' Yrarhaok, Nutionni rounril of Tearhers of Mathrmutira p. WO.

9* THE SIXTH YEARBOOK

training have been already so carefully covered by Dr. Reeve,by the writer, and by others, that extended additional diseuuionhere is unnecessary. It may be well, however, to place the em-Oasis of repetition on the following basic considerations,

The number of fundamental propositions demonstrated in thetext should be reduced to the minimum absolutely necessary as aframework for the subject.

Original demonstrations should be encouraged and simplicityand conciseness the .ein should be insisted on.

No sharp line should be drawn between plane and solid geom-etry. They can and should be taught in a combined course thatneed not exceed one school year in length.

It has never been well demonstrated that trigonometry belongsin the high school curriculum, particularly in the college prepara-tory courses. If taught there at all it should be limited to thosesimpler phases directly applicable to the solution of practicalproblems and its presentation should be made a part of the workin plane geometry. It is not that elementary numerical trigo-nometry is too difficult to be grasped by the mind of the highschool student but rather that, as a rule, when taken as a completesubject in the high school, plane trigonometry is seldom coveredin its entirety. The student having been once exposed to it inhis high school experience generally passes by the college coursein trigonometry with its broader applications and point of viewwith the result that he is usually lame throughout his study ofanalytic geometry 1.tal the calculus, even though by the time he hascompleted thssse subjects he may have attained a mastery thatenables him to handle still more advanced phases of mathematicswith credit.

The value of practice in application of geometry to the solutionof practical problems cannot be overemphasized, both as a stimulusto the interest of the student and as a valuable lesson in the prac-tical utility of fundamental science in everyday affairs.

Calculus in High School Courses. Right here it is of interestto note the discussion by Dr. Peeve of "Calculus in the HighSL liool," together with his inclusion of elementary calculus in theoutline of a prospective course in mathematics for twelfth gradesenior high school) students.' This is perhaps a hold step, but

t uurth Yearbook, Sational Couneil of l'earker4 of Mathematiem, pp. 1114 and173

MATHEMATICS IN AGRICULTURE

within the limits of available time it must be conceded to be allow-able for the idea and the use of the first derivative, for example,are not difficult to grasp, and experience has shown that it fre-quently clarifies ideas commonly included in the elementary mathe-matics of high school curricula which the students find difficult tohandle or to understand fully without the aid of the derivative.The subject of maxima and minima is a clear case in point.

The writer cannot agree with the idea that daily work by theclass at the blackboard is a mistake. Pumice of the blackboardtype with a well-organised class, when consistently followed in awell-planned course in geometry and supplemented by rigid ques-tioning and criticism by both class and teacher, uncovers and breaksdown the bud habit of memorising prooft, stimulates keenness ofinsight, power to visualise, and clarity of logic. The best resultsin the teaching of geometry have followed systematic daily boardwork, broken occasionally by special presentation of originals orparticularly complicated regular theorems by selected or by volun-teer members of the class. if any great amount of tune is wastedin connection with board work it is unquestionably due to lackof system and class organization for which the teacher alone isresponsible.

VALVE OF l'xrry AND THE UTILITARIAN POINT OF VIEW

In closing this discussion it scents desirable, even at the risk ofsome repetition, to outline a certain broad foundation on whichto base the requirements in mathematical study for any field ofapplied science. This, of course, includes modern agriculture,whether considered directly or through the avenues of its manyclosely related fields of interest.

Mathematics Inseparable from Human Experience. It. hasakays been difficult to understand the antagonism displayed bymany people to the idea of arc wiring mathematical knowledge andunderstanding; for, whether or not one recognizes or acknowledgesthe fact, we live in a scientific cosmos in which the fundamental,governing foi''es all act in accordance with defin;te law. Law is theidea, whether expressed or only implied, of the method of orderlyprogress, and orderly progress is mathematical in its very essence.Home, no one can escape the continual influence of mathematicalprinciple in his life (verience, no matter how much he may claimor seek to do so.


Unity of the Science the Key to Success. However, mathe-maticians and other scientists should recognise that, among a lammass of people, they have to meet this antagonistic state of mind.It seems reasonable to assume that the simpler, the more unified,and the more directly applicable to the problems and interests ofdaily life the science of mathematics can be made to appear to thepublic mind, the more successful will they be in overcoming thisantagonism,

We must, therefore, think of mathematics, not us arithmetic,trigonometry, or calculus, not as algebra, frequency distribution,or theoretical mechanics, but us the science of quantity and itsrelationships to thought and tuition with all their material accom-paniments, In other words, in our general consideration of thescience of mathematics in relation to human life, education, andprogress, we must wipe out the rigid artificial boundaries set upthrough centuries of stilted and pedantic scholasticism betweenwhat for want of a better term we are wont to call "the branchesof mathematics," for no such lines of cleavage exist in scientific fact,

Utility Demanded by the Youthful Mind, One of the essen-tials in mathematical training that applies with especial force tothe broad field of agriculture is the development of its eminentlyutilitarian aspects, That is, we must seek continually to developthe ability of the student to apply mathematical principles to thephase of human interest.

To accomplish this end in these times of swift progress, whereinboth physical and intellectual inertia arc being inure fully uvertouwalmovt dull) by new and startling developments that have nearlyeliminated time and space in human experience, we must be ready

enter into the of present-day youth. This means thatwe must always be On the alert to secure rya/ peoblems fromany and every part of the field of activity. We must present themto our students not alone for practice in their solutionwhichmust be insisted uponbut even more to arouse their interest andto show the utilitarian side of mathematics in practically everyphase of human interest.

Old Type Problems Obsolete. The youth of to-day fromkindergarten to college commencement is little interested hi theinfantile, senseless, or purely artificial problems with which ourmathenutties texts from arithmetic to calculus have been wont tobe packed. Rather is he interested in what he sees actually going

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MATHEMATICS IN PHARMACY AND INALLIED PROFESSIONS *

BY BDWABI) SPEASESchool of Pharmacy, Western Reserve University,

Cleveland, Ohio

Introduction. The Yearbook is doubtless the proper placefor a presentation of the usefulness of mathematics in pharmacy,medicine, dentistry, and nursing, inasmuch as it will be read bythose most interested in the preparation of young men and womenwho are to enter these professions.

This chapter will not deal with mathematics needed by the re-search worker in the professions mentioned, but will confine itselfto the mathematics useful to the study and practices of the pro-fessions.

The Mathematics Curriculum in the Professional Schools.It may be of interest in the beginning to state that little or nomathematics, as such, is taught to the medical student after heenters the medical college, for it is presumed that he is properlyprepared to carry out all necessary calculations confronting himin his work. The dental student in some instances is taught someapplication of mathematics in his course in materia medica. Thenursing student is taught some mathematics in the course knownas "Drugs and Solutions." Nearly every pharmacy school to-dayoffers a course in pharmaceutical mathematics. The teaching ofapplied mathematics in these curricula which has come aboutduring recent years is due, I believe, rather to the growing con-sciousness of the importance of the subject than i'ierely to theunpreparedness of the entering students.

It. would be an easy matter to shift the responsibility for thelack of knowledge of simple arithmetic and algebra to the highschool and grade school teachers; but is it not more reasonable toassume that the student entering a profession will probably be more

Readers who are interested in finding further problems of tile kind hereinillseussed should consult Ph artnareutfral Atathytnattra by the author of tidoarticle and published by Mairan,11111,---Ttig Boma.102

PHARMACY AND ALLIED PROFESSIONS 103

or less unskilled in the mathematics of his childhood and its appli-cation in the professions? Should we not rather expect him tohe taught the application when he readies the study of the pro-fession, and to have not only ability but also alertness enough toreview what he most needs?

Nearly every student entering a medical eollege to-day willhave had at least one year of college mathematics, end many ofthem will have had some calculus. This will be true also of thestudent of pharmacy before he has completed his college course,as most of the collek,es of pharmacy give at least one year ofnuithemties in the four-year course. This year will enibravecollege algebra, trigonometry, and analytical geometry. Somecolleges restrict this year to the first two subjects mentioned andsome to the last two.

The four-year course will be the minimum one for AssociationSehools of Pharmacy after 1932. Such mathematics, especially inuniversity schools or where a connection with a college of liberalarts is possible, is taught by a teacher of mathematics or is givenin the regularly prescribed courses with students of liberal arts,sciences, or engineering.

In addition to the above mathematics, all student; of chemistrymust know and use the applied mathematics of chemistry. Themathematics of the freshman year of chemistry is comparativelysimple, but the student of the sophomore or junior year will take acourse in chemical problems, either as a separate course or inter-woven into the course in quantitative chemistry. Most of the stu-dents of medicine and dentistry will have this work before enteringupon purely professional studies, though quantitative chemistry isnot in all instances required of them ; and all students of pharmacywill have it early in their professional work. The diflieult featuresof such a course .ire seldom found in the pure mathematics, butmore often in the application to the field of chemistry.

It appears to-day that the weakness of the students coming toIL: is in simple arithmetic and in very simple algebra, lint it is alsotrue that it is easier to teach the applied mathematics of chemistryand pharmacy to students to-day than it was fifteen years ago.The reader may draw his own inferences.

To summarize what has gone before and up to this point onemay say that the student of medicine, dentistry, and pharmacywill have had the mathematics of grade school and high school and


at least college algebra, trigonometry, and analytical geometry,and some will have had both differential and integral calculus.The student of nursing entering upon the combined courses nowoffered in some universities will in some instances take college alge-bra and trigonometry.

Prerequisites in Mathematics. Class A medical schools re-quire two years of liberal arts work for entrance, and althoughmathematics is not required in most schools, it is suggested as be-ing a useful background course. Some medical schools require threeyears of liberal arts work and others four. The four-year coursein pharmacy in nearly all instances embraces the study of at leastone year of college mathematics exclusive of the applied mathe-matics of pharmacy and chemistry.

Any student who is not properly prepared in mathematicswill experience difficulty in any of these four professions. Thebest advice that a high school teacher can give to a boy or girlplanning to enter any one of these professions is to take all themathematics his high school offers. I do not say definitely that astddent who has not had all the mathematics available in a highschool will fail in one of these professions. I do say that I havenever known one who has had such work to fail and I do knowthat the student who is good in mathematics and who has takenmuch of it finds his professional work far easier.

Applied Mathematics of Pharmacy and the Allied Profes-sions. The applied mathematics of the four professions is verysimilar, though perhaps the pharmacist finds a range greater thanthat of the others; so the subject from this point on will be dis-cussed from the basis of the elementary mathematics employed andits application. No attempt will be made to classify it for each orany one of these professions. The above statement is of coursemade without reference to the mathematics necessary for the re-search worker.

First are taught tables of weight and measure in common useand how to change denominations from one table into like denomi-nations in another. A student of any of these professions or astudent of chemistry who cannot understand and make these trans-positions easily and quickly is handicapped. The metric systemis now accented but the student must be familiar with the avoirdu-pois, troy, and apothecaries' weights as well. In pharmacy, labora-tory work accompanies or follows this study and practically all


calculations found in the mathematics course are carried out inthe laboratory.

The writer has seen the following experiment made with studentsof medicine and of dentistry and the interest displayed by themwas positively amazing. Weigh a grain of wheat, using apothe-caries' weights, and then weigh it upon a fine analytical balance,using metric weights. The fact that one grain is equivalent tonearly 65 milligrams has taught some concept. both of the grain andit.3 origin and of the size of the Milligram. I shall leave it to thereader to decide where this instruction belongs, whether in gradeschool or in college. I believe I can argue upon either side of thequestion but all will surely agree that the student of these profes-sions needs the knowledge. It may be an innocent sport, if onehe interested in this subject, to ask. his physician, his dentist, hispharmacist, or his nurse as he meets them, What is a grain? Whatis a milligram? The writer once had a class of fifty freshmen,twenty-two of whom had never seen a grain of wheat.

In addition to an understanding of the tables of weights andmeasures and the transposition referred to, it becomes necessary forthe student to develop practical knowledge so that answers to givenproblems are weighable and measurable with the usual apparatusat hand. This is a practice that may well start in the early studyof arithmetic. A result involving a weight, volume, or linear meas-ure should always be stated in such terms or denominations thatit is of practical usefulness.

As an example of an improper answer for a problem I presentone which I once received from a state department. I had askedfor a definite location of a culvert on a canal. The map showedit to be a short distance from a bridge and also a short distancefrom a canal. called a "side cut." leaving the main body of water.The answer given me was somewhat like this: "446,700 feet fromthe north corporation line of " (a town several countiesaway). The point I wished to locate was just a trifle more thana mile from the "side cut."

The review of ratio and proportion is always necessary, as itis a short cut to the solution of many everyday problems. Theterm "ratio" and the mathematical expression of it not only arefound in our daily labors .ry work but will be found in a vastnumber of the texts and Arth.'es read by the four professions ineveryday life.


I am not familiar with the method employed in preparatoryschools to teach this subject, and I sympathize with the teacherwho must teach it; for I find there are some students who cannever really grasp it. simple though it is. Students who come tous from city schools seem fairly familiar with the fractional methodof expressing a ratio, but have not been taught the real meaning ofa proportion or that it may he stated in whole numbers as well asin fractional form. Ratio and proportion are used in both chemi-cal and pharmaceu+ical calculations and are well-nigh indis-pensable.

The reading and writing of fractions and the transposition ofthe common fraction to the decimal and vice versa are almost dailyoccurrences.

To the student who cannot. master relative sizes, volumes.weights, and other dimensions and who does not have a mental pic-ture. of fundamental units, such things as the capacity of bottlesor laboratory glassware constitute a never-ending problem.

An examination was given to a number of recent high schoolgraduates. It may be of interest to observe that they came fromlarge city, small city, and small town high schools. They were'given a problem which is not an uncommon one in everyday prac-ticeto calculate the volume in gallons of a drum (eylinde-i.The figure of 231 cubic inches in a gallon was furnished, but notthe rule. They all knew the rule and how to apply it. Thedimensions of the drum were stated to be 36 inches in height, and18 inches in diameter. One student's answer was "Four gallons."He had made a mistake in calculation but did not observe thatalmost anyone could hazard a guess or make an estimate thatwould he closer to the real answer.

Some practice. in addition to that of actually proving re-sults, should be given to the end that a student will not present aresult that is ridiculous. It is needless to offer more than theabove suggestion to an intelligent teacher as to the field open inthe laboratory for the use of mathematics.

The subject of percentage is very important and it likewise isa much abused subject in these professions. We use not only truepercentage but approximate percentages and near percentages;and therefore, unless its real meaning be perfectly clear to thestudent, much confusion results. It is very doubtful if the studentexists who cannot calculate 3r,`, or 6 ; of any given number of


dollars or any given number of pounds, but the converse of evensuch a simple problem always seems difficult. The application ofpercentage to business problems, to percentage strength of com-pounds and to the mixing of substances of different strengths, tothe dilution of substances to a definite percentage strength, and tothe fortification of others always seems to mean something dif-ferent from the old friend, percentage, of grade school days. Thereare, of course, some things that interfere with the mixing of sub-stances of different percentage strengths that complicate what isotherwise a simple mathematics problem. For example, one hun-dred parts by volume of 95r,; by volume alcohol mixed with 100parts by volume of water will only make about 190 parts by volumeinstead of 200 parts. Such procedures certainly should be taughtin an applied course and not in a preliminary course confined topure mathematics. It would he helpful, however, if teachers couldemploy percentage in other ways than the relationship of it tofinancial considerations. Why not study ratio and proportion inmixtures such as concrete, one cement and three sand, or cement,sand, and crushed stone, and then express them in terms of per-centage? How about volatile matter and ash from coal? Suchexamples would help us in percentage purity of chemicals, per-centage composition of compounds, and percentage strength of so-lutions.

Under the heading of the calculation of dosage we make use ofboth common and decimal fractions and it is often necessary toadd. subtract, multiply, and divide them. The average studenthas usually forgotten iiow. but it does not take long to bring backthis knowledge, and my observation is that once back, it stays,because it is in daily use.

A nurse might be confronted with the problem of giving a 1/10grain dose of a substance and all that she has are tablets of thesubstance, each containing 1/4 of a grain. Fractions first, ofcourse, and then knowledge that a tablet cannot be divided withsafety, that small amounts cannot be weighed accurately upon theapparatus at her disposal, and so on ad infinitum.

If time is not too short the student should be taught why% % = 7/12 and why the common denominator is used. I heremean to express that the student knows how to make such cal-culations if he has not forgotten mechanical instructions, but hedoes not seem able to see practical everyday problems and usage.


He views these calculations as school tasks of to-day and does notsee their relationship to life problems.

When it comes to calculating specific gravities of solids andliquids, the laboratory helps the student's mathematics. The old-fashioned drill of giving two factors in a three-factor product tofind the third often simplifies the entire teaching of this subject.After the student learns what specific gravity is and begins to useit, his mathematics becomes clear. This is a subject which he hasusually had in high school physics or in grade school mathematics,and sometimes even in college physics, but its value has alwaysbeen academic to him.

Calculation of allowable errors in weighing and measuring isalways more or less difficult. This again may often be compli-cated by percentage. Suppose a student is told to prepare five. gal-lons of a mixture so that one teaspoonful will contain of agrain of strychnine sulphate. How accurate must be the balanceupon which the strychnine is weighed, and what percentage oferror in either direction does his good judgment tell him is allow-able? Here again is applied mathematics. Percentage solutions,solutions made on a ratio basis, and saturated solutions all involvesomething besides mathematics, but the simple mathematics mustbe known first. The difficulties usually involved in teaching con-version of temperature scale readings are obviated by teaching thesereadings as definite measurements of length. Interpolation in spe-cific gravity tables becomes easy to the student who has usedlogarithms and this last subject is one that is needed by the studentof medicine and pharmacy very often indeeti. Old-time alligationhas to be taught pharmacy students because State Boards demandit. The writer, however, is not an advocate of its use.

A student always needs to understand problems of interest,discount, profit and loss, and many other problems found in theaverage commercial arithmetic.

The Preparation Students Should Have. The writer feelsthat he will not do justice to himself unless in elosinr lie stateshis own opinion upon the preparation of students in mathematicsin order to make professional work easier. That opinion is thatarithmetic should continue into the high schools. either in the formof the old-fashioned advanced arithmetic or in the form of reviewin the senior year. I make this statement knowing full well theobjections I should hear if I happened to he a well-known teacher


of mathematics. I hasten to add, however, that arithmetic shouldnot replace the algebra, geometry, and in some instances highermathematics now taught; it should be taught in addition to them.A student expecting to study any one of the four professions which

we have discussed, or one expecting to study chemistry or physics,needs mathematics every day.

Much of the necessary arithmetic: may be reviewed by theteacher of algebra. and may I add here that the teacher of algebrashould endeavor at all times to show, where possible, that muchof algebra is really a review of arithmetic and is after all an easiermethod for solving many simple problems. The simple equation,to find the value of x, still seems to he difficult for many studentsto solve. They still find it difficult to form a simple equation andsolve it, and for such studoits the use of the simultaneous equationis out of the question.

In my own institution we were once confronted with the prob-lem that our required course, embracing mathematics, was too se-vere; that students who did not wish to become scientists and realprofessional men should he permitted to take economics insteadof mathematic and language. It was their desire to know merelyenough of science to he safe technicians, and otherwise they wouldhave purely business interests. It soon developed that those whocould not pass in mathematics felt they could join this secondclass, and so our poor students went to the economics classes and(lid not endear us to the professor of economics. Before long wediscontinued this practice, for we arc of the opinion that businessas well as science needs men of brains who have a working knowl-edge of mathematics.

I must plead guilty to belonging to that class of old-fashionedpeople who, if given the chance, would require for high schoolgraduation arithmetic., algebra (three semesters), plane and solidgeometry, and as much more a could be put into the course. Someof the writers upon mathematics have felt that a knowledge ofmathematics and an ability to think do not necessarily go handin hand and that the first. does not tend to develop the latter; butthose of us at this end of the scale are usually fairly certain thatwhen we get a student who is a good student of mathematics andhas been well grounded in the subject, most of our troubles fadeinto nothingness so far as he is concerned. May I add that I do notapprove of applied subjects of any kind until after the funda-


mentals have been taught? There is always the danger of slightingthe fundamental for the more interesting application.

I wish the specialist, the real mathematics teacher, to give thestudent his elementary training. I know the specialist will do thispart of the work well, and I can then find the means to teach thestudent the practical and professional application of the mathe-matics he has learned.

MATHEMATICS AND STATISTICSfly HELEN M. WALKER

Teachers College, Columbia University, New York City

1. STATISTICAL METHOD IN PRESENT-DAY THINKING

Importance of Statistical Method, More and more themodern temper relies upon statistical method in its attempts tounderstand and to chart the workings of the world in which welive. Particularly in those sciences which deal with human beings,whether in their physical and biological aspects or in their social,economic, and psychological relations, the spirit of our time asksthat its conclusions be based not so much upon the distinctive re-actions of one or two individuals as upon the observation of largenumbers of individuals, the measurement of their common likenessesand the extent of their diversity. As the data thus gathered frommass phenomena become extensive, it becomes imperative to havemethods of organization to bring the facts within the compassof our understanding, methods of analysis to make the essentialrelations appear out of the mass of detail in which they are hid-den, and methods of. classification and description to facilitate thepresentation of the data for the study and consideration of otherpersons. Thus statistical method becomes a telescope through whichwe can study a larger terrain than would be accessible to ourunaided vision.

Use of Numerical Data. As the area of investigation is wid-ened to include larger masses of individuals and as the nature ofthe inquiry becomes more precise, it is inevitable that data andconclusions shall assume numerical form. To quote Sir FrancisCalton:

General impressions are never to be trusted. Unfortunately when theyare of long standing they become fixed rules of life, and assume a prescriptiveright not to be questioned. Consequently those who are not accustomed tooriginal inquiry entertain a hatred and a horror of statistics. They cannotendure the idea of submitting their sacred impressions to cold-blooded veri-fication. But it is the triumph of scientific men to rise superior to suchsuperstitions, to devise tests by which the value of beliefs may be ascertained,

111


and cu feel sufficiently masters of themselves to discard contemptuouslywhatever may be found untrue,

Sir Arthur Newsholme writes:As the scupe of a science widens, it is generally found necessary sooner

or later to adopt numerical t'tandards of comparison. In medical sciencethis is found to be especially necessary, though perhaps in no other scienceis the difficulty of exact numerical statement so great. The value of experi-ence, founded on an accumulation of individual facts, varies greatly accordingto the character of the observer, As Dr. Guy has put it "The sometimes ofthe cautious is the often of the sanguine, the always of the empiric, and thenever of the sceptic; while the numbers 1, 10, 100, and 1,000 have the samemeaning for all mankind."

Adolphe Quetelet, the great Belgian astronomer, mathemati-cian, anthropometrist, economist, and statistician, in the first lec-ture of a course on the history of science, said:

The more advanced the sciences have become, the more they have tendedto enter the domain of mathematics, which is a sort of center towards whichthey converge. We can judge of the perfection to which a science has comeby the facility, more or less great, by which it can be approached by calcu-lation.

Relation of Statistical Method and Statistical Theory.Clearly, then, statistical method must be grounded in statisticaltheory, which is essentially a branch of mathematics. Indeed, sta-tistical theory has its roots in the mathematical theory of prob-ability and the work of the mathematical astronomers, notablyGauss and Laplace. who early in the nineteenth century built upa theory of errors of observation in the physical sciences. Statis-tical method is completely dependent on statistical theory, yet thetwo have important differences in purpose, in procedure, in tech-nique, and in the type of talent and preparation needed for suc-cessful prosecution.

Statistical theory is developed for an ideal situation seldomcompletely realized in practice. Statistical method almost alwaysinvolves a measure of compromise between the recalcitrant factswhich life presents for analysis and a mathematical theory whichpostulates a particular form of distribution or other ideal circum-stances only approximated by the data. The dependable statisti-cian recognizes that the assumptions implicit in his formulas arenot completely fulfilled. but lie secs that to use these formulasand continue the investigation will afford a far closer approach to

MATHEMATICS AND STATISTICS 113

the truth he seeks than mere conjecture and intuitio:., which maybe the only alternative.

Difference Between Statistical Method aad Mathematics,The man who develops a new piece of statistical theory works asa umthematician and faces only those obligations ordinarily incum-bent upon the mathematician. He must 6tate his assumptions, hemust avoid contradictory assumptions, he must be careful thathis conclusions follow logically from his premises. I3eyond thatpoint, he is free to make whatever assumptions are convenientfor the simplification of his argument. The worker in statisticalmethod who applies to the solution of some practical problem aformula thus developed by the mathematician has an additionalobligation. He must not only know the 1188111111)001IS on whichthe formula rests, but he must also b now the content of the fieldin which he is working well enough to determine whether theseassumptions can be appropriately made in that particular situation.It is probably not essential that he be able to go through the stepsof the derivation of each formula, but unless he knows what as-sumptions were made in that derivation and unless he ascertainsthat these assumptions can be reasonably made for his data, thereis a possibility that he may come out with conclusions which arefar from correct. Herein lies the weakness of many statisticalinvestigations, either that the research worker does not know themathematical theory well enough to recognize the assumptionsupon which his procedure rests, or that he is not sufficiently athome in the field of research to pass upon the validity of thosehypotheses,

Knowledge of statistical theory is not enough for the man whowould plan important statistical investigations. Neither the puremathematician nor the man innocent of mathematical trainingmakes the best worker in practical statistics. The expert in sta-tistical theory needs also a rich knowledge of the content of thefield in which he would work. The general method of statisticsis the same for all fields and the elementary training need notdiffer much whether a man is to work in biology or psychology,in economics or education. Therefore it is sometimes suggested thata consulting statistician trained primarily in pure mathematics andin statistical theory may act as consultant for a large number ofimportant statistical studies in various fields, the data for whichare collected by others and the results worked out by others. By


providing expert advice on the method of research, sorb a manmight make it possible for important studies to 1w milled out bymen interested in the content but ignorant of statistical method.

This solution has serious drawbaeks. Without a general knowl-edge of the field of study, it is difficult to choose appropriate statis-tical procedure. Without a consuming interest in the outcome ofthe particular study and an intimate knowledge of its details, fruit-ful leads do not arise, "hunches" are lacking, and the most signifi-cat facts and relations may be overlooked. Recently a researchworker in biology wrote to ask me if he was justified in using acertain procedure. Not being a biologist I could offer no creativesuggestions, and could only say, "The assumptions underlying theformula you mention are thus and so. I should be suspicious ofthem, but a biologist will have to pass upon their applicability,"

On the other hand, the view is common that the mathematicianwho develops a formula has welded a tool which the tionmathe-matical psychologist or economist or educator can profitably usewithout knowledge of its derivation, and that the formulas printedin the textbooks constitute a dependable machine into which datamay be fed and from which conclusions, even discoveries of vastmoment for human welfare, will eventuate automatically. Twoboys on the top of a Fifth Avenue bus arrested my attention witha scrap of corm.rsation. Said the first, "But I don't understand

; whereupun his companion replied, "Understand it? Gosh,n an, why should you try to? It's a formula!" The world is w1of men who want to take formulas on faith, arguing that they canutilize a icelmique whose basis they do not understand, exactlyas th..y drive an automobile which they could not take to piecesand reconstruct. The analogy has something to recommend it.Certainly, most of the computation and tabulation vaned for in astatistical study can be done by clerks wl:o merely follow diree-tins. For the man who is directing ruearch, however, the analogytails. In driving a ear we have a perfect nni obvious check uponthe success with which gears and steering wheel are managed. Inchoosing a statistical procedure, no such obvious check is available.A formula is always based upon assunptions made during theprocess of derivation and these assumptions limit its application.The formulas printed in texts are often special cases of longer onesan deduced from them by the application of very special as-sumptions. The choice of a different formula may vitally affect

MAIIIEMATICS AND STATISTICS 116

the nature of tads results, but there is usually nothing in the re-sults themselves to validate the method by which they werereached,

The Nature of Statistical Inference, Statistical reasoningdiffers train mathematical reasoning in another important way,For the mathematician, conclusions follow inexorably and inevitably from premises; his reasoning eventuates in a single im-mutable law which is always invariably true when the original con-ditions prevail, The statistician derives no law which can applyinvariably to all members of the population which he studies; buthe deduces trends, tendencies, which are true in the main for thegroup, but which may not hold at all for a given individual, Hespeaks of the central tendency of the group and of the tendencyof the group to depart therefrom, of the scattering or divergenceof the group from that central tendency, The mathematicianknows all his premises; the statistician can usually measure onlypart of the influences which play upon the subjects he is s tudying.The statistician is usually working in a field where events arebrought about by a highly complicated plexus of muses only 'milof which can be measured, and therefore he speaks of probabilityrather than certainty, He recognizes that when he is studying onehundred cases and is attempting to generalize for ten thousandcases the results which he obtained from the hundred, then everymeasure he has computed fur the hundred probably differs a littlefrom what he would find if he computed the same measure furthe ten thousand. Such samIding errors cannot possibly be avoidedand can be mitigated only by the use of larger samples. When amathematician speaks of an error, he means a mistake. When astatistician speaks of a sampling error, or when he computes aprobable error in the attempt to measure the signifivnce of hissampling error, he is nut dealing with a mistake but with a funda-ental characteristic of the nature of the universe which makes

one sample differ slightly from another.It is the very essence of statistical method that it describes the

trends and the general characteristics of pipulatios, but that thesetendencies cannot he asserted us necessarily valid for each of theindividuals which constitute the group. To describe these tenden-cies and relations with objectivity and precision, quantitative andnumerical measures are naturally called for. If we say, "Mostof the teachers in time country schools of America receive very low

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think statistically in order to read his newspaper and the currentmagazines.

Suppose, for example, that. a nutrition expert interested in thefeeding of small children wants to find out whether two-year-oldshave their appetites stimulated by the color of the food which isoffered them, and plans to make daily observations of fifty nurseryschool children. His training is in child psychology and foodchemistry, but as the investigation progresses he finds himselfinvolved in a statistical study of some complexity. Or again, ateacher of physical education wants to arrive at an index ofphysical capacity which will allow him to estimate in advancethe ease with which a given high school student can master aparticular sport, say swimming. His interests are in anatomy andphysical training, while this problem is largely one of appliedmathematics.

It, Iris introduction to Wood's Measurement in Higher Educa-tion f lt)231, Professor Louis Ternian says:

prom the language of statistics there is no escape if we wish to go beyondthe limits of p,rsonal opinion and individual bias. Worthwhile evaluationsin higher e,11.ration will continue to be ns rare as they now unhappily areuntil the rank and the of college and university teachers become able tothink in more exact quantitative terms than they are yet accustomed to.

More than a quarter of a et :dury ago 11. (;. Wells said:'bite to tr ma thelllatieS is a sort of supplement to languge, atTording a

means of thought about form and quantity and a means of expression moreexact, compact, and ready than ordinary language. The great body ofphysical science, a great deal of the essential facts of financial science, andendless social and political problems are only accessible and only thinkableto those who lutve had a sound training in mathematical analysis, and thetune rusty not I u far remote when it will be understood that for complete

o ti as an efficient citizen of one of the great new complex world-widestales that are now developing it is as necossary to be able to compute, to111111k in :11 cragps and maxima and minima as it is now to be able to readand to write. (Mankind in Ike Making, 1904, pp. 191-192.)

The time of which Wells then spoke is now imminent.Mathematical Preparation. How much mathematics should

one know before undertaking the study of statistics? No au-thoritative answer can be given to this question, for no one, sofar as the writer knows, has made a careful, unprejudiced analy-sis to see what mathematical knowledge is needed for variou.4statistical undertakings. If a canvass of expert opinions were


made, they would undoubtedly range all the way from that ofthe mathematician who would insist upon a doctorate in puremathematics to that of the man who once told the writer that heconsidered much study of mathematics an actual detriment, be-cause he thought a statistician was freer and more original if hedid not know too much about mathematics. It is obvious thatthere are many levels at which statistical work is carried on, andit would be valuable to have a thorough study made of the op-timum mathematical preparation for each level. The suggestionswhich follow must be interpreted as based solely on the personalopinion of the writer, which in turn is derived from the observa-tion of students and from the results of a statistical study of thepreparation and accomplishment of over four hundred students ofelementary statistical method.

The clerical worker who merely tabulates and copies raw dataprobably performs no mathematical function at all. The computerwho works under the close supervision of someone else needs tohave a flair for figures, skill on computing machines, a high senseof accuracy and reliability, and enough knowledge of arithmeticand algebra to enable him to see short cuts in arithmetic opera-tions, but he can be a competent worker on this level with relativelylittle theoretical training. He can make extensive computationsunder direction without much understanding of the import of hiswork.

The student who hopes to do anything at all with the theoryof statistics should have, as minhnum preparation, differential cal-culus. While it is true that Yule wrote his Introduction to theTheory of Statistics without any of the notation of the calculus,nevertheless he could not avoid its general method, and most ofhis readers will agree that he did not succeed in simplifying hismaterial by this expedient. The student who goes beyond the firststages in his study of statistical theory and who attempts to readthe original memoirs in which important derivations are set forth,will find that he needs to know integral calculus, differential equa-tions, theory of probability, a great deal about the convergence ofseries, function theoryin fact, almost any form of mathematicalanalysis which he has studied will be of ultimate use. The geome-tries are in general less pertinent, although there arc one or twoimportant papers which have utilized geometric concepts. Theman who is to do original research in statistical theory will be


thankful for all the mathematical analysis he has studied, and willbe exceptional if lie does not feel the urgent need of more trainingthan lie has had in mathematics.

Between these extremes, the clerical worker and the student oftheoretical statistics, there is a large and rapidly growing groupwhose needs are more difficult to define and to meet. These arethe men with primary interest in other fields who need some knowl-edge of statistical method in order to read the technical literatureof their field, to interpret and .evaluate the research of their fel-lows, and to organize data derived from studies of their own. Theyare often impatient with any 'ittempt to teach them statisticaltheory, and they say they are interested "only" in practical in-terpretation and critical evaluation, failing to understand thatcritical evaluation and wise interpretation often call for a fine com-bination of acrinen, wide knowledge of the field of study, and someknowledge of statistical theory. The man who is reduced to quo-tation of what other people have said or written about a formula,having no first-hand knowledge of its bases, may also be limited toimitation and quotation when he attempts to interpret its practi-cal meaning in a concrete case,

However, let us suppose that we are attempting to teach asmuch of statistical method as can be compassed without the deriva-tion of formulas, teaching only statistical computation and asmuch critical interpretation as is intellectually feasible to studentswho make no mathematical derivations. What mathematics isessential to such a program? If we postulate a course in statisticsstripped to a minimum of mathematical content and designed to beof the utmost practical help to time person who has studied nomathematics: beyond the high school, what topics in secondaryschool matheinaties will be most needed? Here again it is neces-sary to make a declaimer and to admit that there is no authorityfor an answer save personal opinion based on a study of the needsand difficulties of a good mummy mature students suffering frommathematical anemia.

The question falls into two parts: Which of time topics nowcommonly taught. in the secondary schools do students of elemen-tary statistics use most? What topics not commonly taught insecondary school mathematics would be useful to the prospectivestudent of statistics, simple enough to be grasped by high schoolpupils, and of sufficient utility for the social sciences to merit


consideration as possible additions to an already crowded cur-riculum?

III. TOPICS IN ARITHMETIC AND ALGEBRA MOST NEEDED PONBEGINNING STATISTICS

Attitudes and Habits. For success in the general processes ofelementary statistical method, it appears that mathematical in-formation and specific skills are less important than certain atti-tudes of mind which are sometimes regarded as by-products. Thatelusive thing which we call mathematical ability seems to be moreessential than mathematical training unless the training producesthese habits of mind. If a student has forgotten how to handleradicals and logarithms he can relearn these techniques easily.If he has never rightly understood the import of a formula, if henever knew what the solution of equations or the reduction offractions were about but merely acquired skill in performing cer-tain tricks which produced an answer, if he thinks variable andunknown to be synonymous terms, if he has never seen arithmeticgeneralized into algebra, these are matters much more seriousthan a total lapse of memory. Worst of all, if lie is unable tothink in terms of symbolism, is frightened by algebraic formulas,panic-stricken when obliged to compute, and without conscience inthe matter of accuracy, he has a heavy load of old habits to dis-card before he can hope for any progress in statistical studies.Fortunately, the ways of thinking which the statistician wouldurge the mathematics teacher to inculcate and develop are waysof thinking which mathematicians also value highly. The fol-lowing are of central importance:

1. Ability to Think in. Terms of Symbolism. The language ofstatistical theory and method is highly symbolic, and no othersingle ability seems so closely related to success in this field asthe ability to read meaning directly from symbolism. In a groupof prognostic tests which we have been giving to students of ele-mentary statistics at the beginning of the first terms work, tt shortsymbolism test including only eleven items shows a correlation of.55 with marks at the end of the first term. For so short a testthis is remarkable. When the symbolism test is made longer,the correlation will undoubtedly be still higher. Of all the otherprognostic tests with which we have been experimenting, none, noteven a standardized test of general intelligence, shows so close a re-


lationship to the term's marks. Even for the pupil who will neverstudy statistics, this ability to use symbolism as a language inwhich to express his ideas is one of the most readily defensibleaims for the teaching of algebra, though not one of the most easilyrealized. Teachers of high school mathematics can scarcely puttoo much emphasis upon translation from symbols to words andfrom words to symbols. In addition to practice in these two formsof translation there should be practice in expressing in symbolicform the pupil's own ideas about quantitative matters, a sort offree symbolic composition. Ninth grade pupils enjoy this whenthe ability has been developed by carefully graded exercises. Itwould be of inestimable value to those who will some day studystatistical method.

2. Correct Thinking About Variables. Hazy thinking whichpermits a pupil to confuse variables with unknowns because bothare often represented by the letters x and y may not be inconsistentwith high marks in a high school algebra course, where it is oftenpossible to achieve correct answers blindly by merely followingthe rules of the game; but such confusion is a very serious handi-cap when algebra is to be applied to statistical method. For ex-ample, suppose we let x, represent the height in inches of one boy,2%, the height of a second, and so on, x being the height of the nthboy. Then the sum of all the heights divided by the number of

2'xboys will be the mean (or average) height, or H N= . Nowclearly x is not an unknown here, for we are not trying to find thevalue of some missing number, but it is a variable representing aclass of numbers to all members of which the formula refers.

In arithmetic a symbol is always associated with the samenumber, 4 having always the same meaning no matter where itoccurs. In algebra the pupil early discovers that x or n (or anyother letter) may have one value in one problem and another valuein another problem; but so long as he is solving such an equationas 3x -I- 2 = 20, x has only one value for that equation. He seesreadily enough that x may mean 6 in this equation while in an-other it may mean 4 or 7 or something else, but he is still essentiallyon the arithmetic level because during the discussion one symbolstands for one number, that number being temporarily unknown.To give the pupil a concept of variables is psychologically moredifficult as well as more stimulating and ultimately more impor-


tant. This is one of the most valuable contributions algebra makesto human thinking and ought to be approached with care andthoroughness. The enormous power of algebra is largely in'ierrntin the fact that a single symbol can be used to represent everyone of a class of numbers. Almost any bright pupil can learn tomanipulate formulas mechanically, to evaluate the formula by sub-stituting given values of the variables, and to change the subjectof the formula; but. unless he has grasped the meaning of a vari-able he cannot think properly about formulas, cannot compose for-mulas to express relations, cannot have any idea of functionality.unfortunately, many pupils can write glibly 13n *1- 7n = 20n with-out realizing that this means, "If seven times any number is addedto thirteen times that same number, the result will always betwenty times the original number." If anyone doubts this state-ment. it is only necessary to ask a class to find the value of13.X 43 7 X 43. and to hand in all their scratch N% ork, and thento note how few think of multiplying 43 X 20 and how many makethe two separate multiplications and add the results.

In statistics we deal constantly with variables, while onlyseldom do we solve equations to discover the value of unknowns,and the student. who has learned to think of x as standing alwaysfor a single missing number has a mental handicap to overcome.

3. Freedom from Fear. Among mature, educated men andwomen, graduate students with intelligence well above average,there exists a surprising amount of fear of anything savoring ofarithmetic or algebra, old inhibitions which are rooted in arith-metic failli. -s and worries in the early grades. old strains andanxieties which can often be traced back to a teacher who wasscornful when answers did not come out right, or who tried tohurry children beyond their capabilities. It is usually a revela-tion to the student to discover that a considerable part of the fearand worry which he had been attributing to the difficulties of sta-tistics actually had their origin in early misadventures with arith-metic or algebra, and when to this discovery he adds the discoverythat a little well-directed practice will rid him of his sense ofmathematical inferiority, he achieves a joyous freedom. But oughtany subject to have such serious emotional connotations amongmen and women who are otherwise sensible and intelligent? Willthe boys and girls who are being taught arithmetic, algebra, andgeometry to-day have to carry a similar load of emotional condi-


tioning toward mathematics? Have we even yet learned to adaptour teaching to the pupil so that he may work up to his individualcapacity without being constantly humiliated and scarred by fail-ure to do successfully work which is beyond his capacity? Can welearn to teach so that boys and girls may have the triumph ofbecoming mathematical adventurers and discoverers and so se-cure a self-confidence which will be a positive force in buildingtheir personalities?

4. Right Attitudes Toward the Outcome of Computation. Thesecondary school pupil usually considers his computations, hisalgebraic manipulation, his geometric reasoning, and his trigo-nometric analysis vindicated by the approval of a teacher or byagreement with published answers. Methods of checking do notinterest hint much; they seem like an unnecessary labor imposedby an exacting taskmaster when comparison with an answer knownto be right offers a more direct proof of his work. In statisticalinvestigations there is no such authority against which one maymeasure his work, and methods of checking become of paramountimportance. The habit of checking each step of a computationor of finding two independent ways to reach the same result mustbe developed in any one who hopes to do valuable statisticalwork,

The habit of estimating in advance of computation what is aprobable vabe for its outcome and of checking each computedvalue by common sense to see if it is reasonable will save thestatistician much grief. Also, time general value of these habitsmakes them worth considerable. attention from the teacher ofmathematics.

A conscience about accuracy is necessary to the good statisti-cian. The correctness of his work is ordinarily taken for grantedby his readers, and only rarely does one man recompute the meas-ures published by another. His work must stand by itself and hemust he able to vouch for its correctness. He should be sensitiveabout the accuracy of his computations, of his tabulations, of themeasurements from which his data were derived. Tie should rec-ognize that the number of decimal places which he carries in hisfinal results is a tacit pledge of the degree of accuracy of his origi-nal measurements, and that he should carry these results only sofar as is appropriate to the precision of the original measurements.He must always realize that unless his problem is a trivial one,


the outcome of his computations may provide information onwhich will be based decisions of importance for human welfare,and that therefore he has no right to offer any but trustworthywork.

What training can we give a high school student to develop inhim this sense of responsibility? Answers are available for check-ing; why bother about excessive accuracy? No social conse-quences wait upon the outcome of the usual algebra problem; aslip in the progress of a geometric proof can bring disaster to nuone unless it be to the perpetrator of the slip. Certainly a desirefor accuracy does not arrive as the result of verbal argument onthe part- of the teacher.

Probably the best expedient is to make one pupil or a smallgroup of pupils responsible for securing data and computing theresults in some matter about which the class wishes to have in-formation. This may approximate the situation of the statistician,whose work has important social consequences and who is stimu-lated by the thought that the outcome may remain unknown unlesshe finds it.

Information and Skills. For a course of the type we are nowpostulating the necessary mathematical techniques are very sim-plcskill and a degree of rapidity in computation, knowledgeof arithmetic short cuts, ability to place a decimal point, to takesquare root, to read a mathematical table, to change the subject ofa formula, to evaluate a formula, to transform fractions, to op-erate with complex fractions, to plot points on coordinate axes, tomake statistical graphs and to interpret them, to draw the graph ofa linear equation and to know the import of the slope of a line,and to handle radicals. The use of a slide rule, of computing ma-chines and of logarithms is highly desirable, as is also the abilityto interpolate.

The formula given below indicates the complexity of structureto be encountered in computing a coefficient of correlation;

a be

e

N2 I` v -One should he able to change the form of this fraction as con-venient and to know enough about radicals to deal with the de-


nominator. Again, it is necessary to evaluate expressions of theforms

v 1 (1 a2) (1 b2) (1 c2),

a bea

bV 1 e2 V 1 (12and

cV1 V1 g2 V 1 b2 V 1 c2.

These suggest about the limit of algebraic difficulty likely to beencountered in a first-year course in which formulas are notderived.

IV. STATISTICAL METHOD FOR HIGH SCHOOL STUDENTS

A Challenge to Teachers. Any one vitally concerned withthe teaching of high school pupils and observant of the rapidlygrowing public need for some knowledge of quantitative method insocial problems must be asking what portions of statistical methodcan be brought within the comprehension of high school boys andgirls, and in what way these can best be presented to them. Ifsome aspects of statistical method are to be taught in high school,shall this be done by the mathematics teachers or by the socialscience teachers? Shall a new course be created, shall a new unitbe added to the social science work, or a new unit be added to thework in mathematics? Shall it be required or elective, for seniorsor underclassmen?

These questions call for much study and creative teaching onthe part of high school teachers with pioneering spirit, and thereseems every reason to expect that the next decade may producesignificant changes in the program of both high schools and col-leges. The situation is full of challenge for those teachers of highschool mathematics who like to leave the beaten path and adven-ture a bit, who are not afraid of the hard study necessary toprepare themselves for teaching in a new field, and who have agenuine interest in that type of social problem which can be ap-proached by a quantitative study of mass phenomena. Suchteachers %rill need first to make themselves thoroughly at homein statistical method, not merely with its elementary phases butwith its spirit and some of its theory. It will be most unfortunateif teachers who have had only a six-weeks' summer course in statis-tical method are the ones who undertake this pioneering. becausethe selection of material for a simple course is not in itself a simple


task, and cannot be well done by the person whose knowledge iselementary.

Suggested Materials. As a starting point for creative andexperimental work in organizing units in statistics to be added tohigh school mathematics courses, the writer suggests a number oftopics which she has taught to ninth grade pupils of average abilitywho found the material interesting, stimulating, and no more diffi-cult than the rest of their ninth grade mathematics.

1. Graphs. Almost any good junior high school text now con-tains valuable material on the statistical as well as the mathemati-cal graph. Special emphasis should be placed on the criticism ofpublished graphs, the analysis of their strong and weak points, andsuggestions of alternate ways in which the same material mightbe presented, with the advantages and disadvantages of each form.Most of the modern junior high school texts now include a treat-ment of the histogram and frequency curve. From these, thecumulative frequency curve follows easily. If the raw frequenciesare turned into per cents and two or more distributions are plottedon the same axes, the resulting diagram provides a way of com-paring two groups which reveals at once many things not dis-cernible from the original distributions. Such diagrams may beused to compare the work of two sections of a class on the sametest, to compare the work of a class with published norms for astandardized test, to compare scores made by boys with thosemade by girls, or to compare the test scores made by a class atthe beginning of a term with the scores of the same class on thesame test at a later date. All of these comparisons are mattersof genuine interest to the class.

For later statistical work, it is valuable to know how to findthe equation for a given line and this is not necessarily a task sodifficult that it must be reserved for college courses in analyticgeometry. Incidentally, this problem has as much intrinsic mathe-matical interest as its converse which is commonly taught, and itcan be treated in a manner simple enough for ninth grade pupils.The pupils may be given a practical problem in measuring anytwo variables that have a linear relationship and plotting the re-sulting pairs of measures on coiirdinate axes. Because of slighterrors of measurementunavoidable inaccuracies which are dueto the fallibility of human eyes and hands and measuring instru-mentsthe resulting pairs of measures will cluster about a straight


line without. actually falling upon it. After the data have beenplotted, the pupil should draw the straight line which he thinksis the best fit for this swarm of points. Then he should find theequation for this line of best fit. Such a problem may be used toopen up a general method of deriving laws which express ten-dencies of physical or social data, to suggest the general methodof curve fitting as employed in the various sciences, to extend andenrich the pupil's understanding of graphs. and to introduce theidea of errors of observation, a concept pregnant with intellectualchallenge if treated by a teacher who has grasped its philosophicalimport.

2. The Percentile System. The simplicity of the percentilescheme (including median. deciles, quartiles, quartile deviation),its frequent use to define the standing of high school or collegestudents on standardized tests, its wide usefulness for describingthe performance of an individual in terms of his position within agroup, the ease with which real problems within the comprehensionof adolescents may be assembled. and the fact that the process ofcomputing a percentile offers an attractive application of per-centage. an introduction to the idea of interpolation and a simpleproblem in intuitive geometry, all make the percentile systemadmirably adapted to the end of the junior high school mathematicscourse. Most of the topics studied in a course in statistical methodare so interwoven that no one of them can be truly understoodwithout knowledge of many others. Because of this interdepend-ence of subject matter there seems to be a fairly circumscribedchoice of topics suitable for the secondary school. The percentilesystem is one of the few topics which can be studied satisfactorilywithout the vexation of continually needing an understanding ofadvanced work to clarify its meaning. This is, moreover, a fieldin which children can be encouraged to produce their own prob-lems. to make measurements and to use the percentile system forreporting results to the class. This furnishes an opportunity forthe pupil to get a VItle taste of the thrill of original research andgives him a simple language in which to report the results of hisOwn independent labors. This is less difficult for the adolescentto understand than some of the problems in financial znathetnaticswhich the junior high school pupil is mastering, and its social usesare no less real.

3. Art.ragc:. The concept of central tendency, or average, is

128 THE SIXTH YEARBOCK

fundamental to the way the modern man thinks about his world.We talk of the average length of life for various occupations, aver-age mileage we get per gallon of gasoline, average sine of classes,average cost of commodities, average age of men and women atmarriage, average salary of high school teachers, average age ofchildren entering high school. It is well for people to know thatthere is more than one method for determining central tendency.more than one kind of average, and that in some situations wherethe arithmetic mean is misleading one of the others may give atruer picture of the group,

The arithmetic mean is easily mastered, likewise the mode. Ifpercentiles have already been studied, the pupils know the medianand are ready now to consider the advantages and disadvantagesof each of these three averages and to discuss which is the best.to use in a given concrete situation. While the harmonic andgeometric means are used somewhat less frequently in practice,they afford attractive illustrations of mathematical principles andprovide excellent applications for work in fractions and in log-arithms, and are probably not too difficult for high school work.(Because 1 have not tried to teach the harmonic and geometricmeans to high school pupils, I hesitate to make a definite recom-mendation.)

The discussion of averages provides an opportunity to clarifyfor the pupil the essential nature of statistical inquiry, to showhim both the importance and the limitations of drawing informa-tion from individual cases, and also the necessity of broadeningthe scope of a study to take in large groups of cases in order togeneralize results of observation. For the sake of a satisfactorylife among his fellows he needs to see that an average is but apartial descripti6n of a group, so that he may not fall into theerror of scorning individuals who deviate from that average. Hecan be shown the need for a measure of the variability of a groupas well as a measure of its central tendency. In the percentilesystEm he has already seen such a measure, and he may be told thatthere is a measure of variability which goes with the arithmeticmean just as the quartile deviation goes with the median, but thatit is a little more difficult to understand and that he will have towait for further work in statistical method to find it.

4. Relationship. The tendency of two traits to be associated.so that .. of them is large the other is likely to be large

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negative correlation between the two abilities, the child who stoodfirst in one would be lost in the other, and the order of excellencewould be exactly reversed for the two lists. Evidently this situa-tion shows neither perfect positive nor perfect negative relation-ship, but there is a general tendency for the people who are highin one trait to be high in the other, and for people who are low inone to be low in the other also Therefore we say that the relation-ship is positive, though not perfect.

We will now draw two parallel lines and will lay off on eachten points equally spaced, as in the diagram on page 131. Be-cause Mary has first rank in computation and fourth rank inproblem solving, we will draw a lir connecting the point 1 onthe computation scale with the point 4 on the problem solvingscale. Because John has second rank in computation and first rankin problem solving, we will draw a line connecting the point 2 onthe computation scale with the point 1 on the problem solving scale.In a similar way lines are drawn to represent the record of eachof the other pupils, one cross line representing the pair of scoresfor one pupil. When correlation is perfect and positive all thecross lines are parallel. The more crisscrossing there is, thelower is the correlation, and the less relationship is there betweenthe two traits.

This form of diagram can be used to greatest advantage witha small numher of cases, say less than twenty-five. The scatterdiagram described later can he used for very large groups.

A class may he divided into small committees, each committeebeing responsible for a report on the relationship between one pairof traits, so that when the charts from all the committees areassembled they will illustrate a number of different problems,


showing varying degrees of relationship, some high and some low.Unless the pupil secs a variety of such diagrams he is likely toattach too much importance to the particular shape of the one hehas drawn.

Psychologically, this use of a graph to portray a statistical re-lationship is widely different from the graph of a mathematicalfunction and should be attempted only by a teacher who has insightinto the nature of statistical inference and who can bridge the

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rather difficult gap between a mathematical function where thereis a perfect correspondence between two variables and a statisticalsituation where the dependence is only partial. Parenthetically itmay be said that the writer is convinced that it is psychologicallyeasier for children to learn the mathematical graph first and thestatistical graph later as an application of the mathematical graph,than to use the statistical graph to pave the way for the mathe-matical graph, as is commonly (lone in junior high school texts.

b) &atter Diagram. Material of immediate interest to theclas3 can be readily found for a problem in plotting a ::catter dia-gram. This is an easy extension of the work in plotting points on


coordinate axes and of the use of a step interval as studied in thedrawing of histograms and frequency curves. It results in a swarmof points as did the experiment described in the discussion of find-ing the equation for a line. When the scatter diagram has beenset up (see any elementary text in statistical method for theprocedure), marginal frequencies should be found and the meanor average for each of the two traits should be computed. Thevalue of each mean should then be indicated on the scale of theappropriate trait, and a line drawn across the diagram at thatpoint. The lines of the two means then divide the area of thescatter diagram into four quadrants. The class may be asked tonote the number of cases which are above the mean in both traits,below in both, or above in one and below in the other, and theymay be told that when most of the cases are either above the meanfor both traits or below for both, the two traits are said to showpositive correlation; when most of the cases are above the meanfor one trait and below it for the other, the correlation is said tobe negative. This is of course a very rough statement, but ninthgrade pupils understand it. They can also understand the generalimport of the appearance o, the mate/ diagram. When the corre-lation is high, the dots tend to cluster closely along a line; whenit is low, they tend to scatter indiscriminately over the diagram.

An exceptionally mature class can go further. They can com-pute the mean for each vertical column in the table, marking itsposition by a small red dot, and then can draw the line of bestfit for these red dots. In the same way they can find the mean ofeach horizontal row, marking its position with a small blue dot,and can draw the line of best fit for the blue dots. These twolines are called regression lines, and they should intersect eachother at the intersection of the means of the two traits mentionedin the preceding paragraph. If we find the slope of the first lineto the horizontal axis and the slope of the second line to the verticalaxis, and if we multiply these two slopes together and take thesquare root of their product, that square root is the coefficient ofcorrelation. This, work would seem to be appropriate only foradvanced pupils in an elective course.

c) Histograms Showing Relationship. The use of histogramsto show the interrelationship of two vulgates may be illustratedby data gathered in our elementary statistics "lasses at TeachersCollege. At the first meeting of the class in the fall a battery of


prognostic tests was given, the results from which were comparedfour months later with the record which the same students madein the first semester's work. Among these tests was one composedof thirty statements purporting to be algebraic identities, and thestudents were told to indicate which of these were true and whichfalse.

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Illack portion represcnts thirty-two students scoring 27 or more. Cross-hatched portion represents forty students scoring 18 or less

Figure 1 is an ordinary histogram showing the distribution ofscores on this trite -false test., the area which represents studentswith scores of 27 or more being black, the area which representsstudents with scores of 18 or less being shaded by crosshatching.There are 32 cases in the black area, 40 in the area shaded bycrosshatching, and 79 in the middle area, which is shaded by widediagonal lines.

Figure 2 is also a histogram showing the distribution of semestergrades inade by these same students in the course in etatistics.


These grades are stated in such a way that the average grade isapproximately 50. From the original data sheet, not reproducedhere, the squares on this second histogram are shaded to correspondto Figure 1. If a student had an algebra score of 27 or more, thearea allotted to him in this new histogram is black. If he had ascore of 18 or less, the area allotted to him is indicated by cross-hatching.

A very close relationship between semester grades and algebrascores would show the black squares all at the upper end of Figure2 and the crosshatched squares all at the lower end. A completelack of relationship would show black, crosshatched, and widediagonal lined squares scattered at random over the area of t'polygon. Colored crayons may he used to advantage. If a morecareful study is desired. the students might be numbered in the

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order of their scores on tlw algebra test. and these numbers writteninto the squares on both diagrams. Then it would be possible tomake a case study of the student who luld a high algebra scorebut made only 36 in semester grade alai of the student with a lowalgebra score who came up to 58 in semester record.

Statements of the following type can he derived from thisgraph: Of the 22 students with lowest semester grades, only one


had a high score on the algebra test. Of the 27 students withhighest semester grades, only one had a very low algebra score.Of the 58 students with semester grades above 52 (more than theupper third). only four i.ad a very low algebra score. Apparentlyknowledge of algebra is not a sufficient condition for high semesterstanding, but it seems to be almost a necessary one.

The Outlook for Instruction in Statistics. At present ourgraduate schools contain hundreds of students of statistics whohave an inadequate background of mathematics. They struggleagainst unnecessary odds because in their high school and earlycollege days no one revealed to them the vital contributions whichmathematics may make to the solution of human problems. Ifthey had seen then that. `the social sciences, mathematically de-veloped, are to be the controlling factors in civilization," as W. F.White has phrased it, they might have elected more mathematicsor they might have approached the mathematics which they tookwith a mind-set which would make it function better when needed.If they had begun the study of statistics earlier in their educationalcareer, there would still be time to acquire the mathematics whichthey need, but making the acquaintance of statistical method onlyafter their graduate work in sortie other field is well advanced, theyfind themselves in a very difficult position.

College courses in statistical inctliod are multiplying with greatrapidity, and the number of students enrolled is multiplying stillmore rapidly. In all probability it will before long become cus-tomary to require an elementary course in statistical method forundergraduates who major in the social sciencesincluding psy-chology, education, and biologyjust as laboratory work is nowrequired of those who major in the physical sciences; and then itwill probably become customary for a course in the mathematicaltheory of statistics to be considered an essential part of the workof a college department of mathematics. When these requirementsare made, certain topics are almost certain to sift down into thework of the high school.

Do the teat hers of high school mathematics wish to leave tothe social science teachers the responsibility for instruction in quan-titative methods of studying mass phenomena?

MATHEMATICS IN PHYSICS *

By IL EMMETT BROWNLittcota SOLout, Teachers College, Columbia University,

New York City

Natural Philosophy. Modern high school physics takes itsorigin from certain courses called natural philosophy which, atleast as early as 1729, began to appear in the academies of Eng-land. By the middle of the eighteenth century, they were BOwell established that several texts had appeared.

Natural philosophy was one of the subjects studied in theacademies in this country from the first. In 1754 we find oneReverend William Smith teaching "natural and moral philosophy"at the "Public': Academy in the City of Philadelphia."' It was apart of the curricula of the English High in Boston (founded 1821)and of the first public high schools in New York (1825).

These courses in natural philosophy were decidedly different incharacter from the college physics of that time and from present-day high school courses. They were descriptive and nonmathe-matical even where the need for mathematical discussion was ap-parently clearly indicated. Largely because of this lack of quan-titative treatment, the texts somewhat resembled those that mightbe used in some of to-day's courses in "Applied Physics." Onetext, Ferguson's, which enjoyed considerable popularity from about1750 to 1825, devoted sixty-two pages to machines and forty topumps. As Mann 2 indicates, these texts were attempting to meetthe demand for secular information whicl. the classics were unableto supply. In many instances, the authors were men whose major

This chapter will deal with high school physics only. inasmuch as a similartreatment for college physic's has already appeared. (See l'ougtion. .1. It., Trainingin High 'School Mathematics Essential for S'arerss in certain ('ollugc subjects.Contributions to Education, No. 403. Bureau of Publications. Teachers College,Columbia University, 1930.) In this discussion, however, not only the m.ithematiesthat is necessary for success in high school physic's will be discussed, but also thegeneral funetion of nuitheinst it's will he emplisiz -i.

+Brown, E. E.. The Making of Our Midtlic .wheels. Longulans, Green & Co.,10(12.

2 Mann, C. IL, The Teaching of Phyortes, p. WI. The Macmillan Co., 1912.136

MATHEMATICS IN PHYSICS 137

interest lay in other directions. Very frequently they were clergy-men. Their purpose was "to bring the rapidly increasing scien-tific knowledge of the times home to young people, without try-ing to force upon them that study of mathematical forms andtheir interrelations which was characteristic of the universityphysics." a The character of these texts and their nonmathemati-cal nature can best be illustrated by the following quotation fromthe 1846 edition of The System of Natural Philosophy, by J. L.Comstock, a physician. (The numbers refer to sections of theoriginal.)

8.5. If a rock is rolled from a steep mountain. its motion is at first slowand gentle, but as it proceeds downwards it moves with perpetually increasedvelocity, seeming to gather fresh speed every moment, until its force is suchthat every obstacle is overcome; trees and rocks are beat from its path,and its rib 'n does not cease until it has rolled to a great distance on theplain.

It is found by experiment that the motion of a falling body is increased,or ace:Aerated, in regular mathematical proportions. . . . It has been ascer-tained by experiment, tl .1+ a body, freely falling, and without resistance,passes through a distance of sixteen feet and one inch during the first secondof time. Leaving out the inch, which is not necessary for our present purpose,the ratio of descent is as follows. . . .

90. If the height through which the body falls in one second be known,the height through which it falls in any proposed time may be computed.For since the height is proportional to the square of the time, the heightthrough which it will fall in two seconds will be faux times that which itfalls through in one second. In three seconds it will fall through nine timesthat space; in bur seconds sixteen times that of the first second; in fiveseconds, twenty-Jive, times, and so on, in this proportion.

Just how far modern physics has departed from the spirit ofsuch writing may he disclosed by a glarce at the portion of anypresent-day text dealirg with this same topic. The factors chieflyresponsible for the character of books such as Comstock's were at.least three in number:

I. The increasingly rapid introduction of machines it,to allbranches of industry with the accompanying demand for moreinformation about these devices.

2. The refusal by the colleges to accept natural philosophy afit subject for college entrance requirements.

3. The belated survival of the naïve, philosophic, nonexperi-mental point of view of mediaeval science.

Mann, C. H.. op. cit.


The Quantitative Nature of Mod !rn High School Physics.It is beyond the scope of this chapter to discuss in detail all theinfluences that have determined the character of present-day highschool physics. However. any attempt to get a perspective of thesubject which failed to take into account the influence of the col-leges would, indeed, be incomplete. This influence was felt in manydirections, and is concealed in many factors which, apparently in-dependent of the university influence, exercised their effect uponthe developing subject of sceordary school physics. In 1872, byrecognizing physics as a subject suitable for college entrance credit,the colleges hastened the disappearance of natural philosophy fromsecondary school curricula. There followed a period in which thedominance of the higher institutions of learning was undisputed.They had set the staop of their approval upon the new subject andinsured its vigorous growth. What. was more natural than for thesubject of physics to acknowledge its fealty and to plan its coursesto be as much like those of its sponsor as possible? Since themodel was the college physics course. a very great emphasis wasplaced upon the standardization of subject matter to this pattern,to the mathematival side of the work, and to a general utilizationof physics for its disciplinary values. The inevitable followed.Enrollment in physics dropped from about 23 per cent of all highschool pupil. in 1895, to 14 per cent in 1915. and to 9 per cent.in 1922. Some of this drop must be discounted as due to the re-moval of the subject from required lists and to the increaseddiversification of the high school offering. However. making allpossible allowance for such factors, it is quite evident that therehas been a real falling off in enrollment in the subje,t, in spite of astrong reorganization movement, which began about 1905, and wasfeatured by the report of the science committee appointed by theNational Education Association.

The Movement Against Mathematics. One of the featuresof this reorganization has been the assault upon the mathemati-cal portions of the subject. There has unquestionably been toogreat an emphasis upon this feature of the work. So long as thedisciplinary theory of eduatitm held, the place of mathematic-was clearly indicated. The more difficult and rigorous the course,the greater the disciplinary value gained by struggling through it.Therefore educ:;tors made the subject more stringent by insertin,great numbers of mathematical problems. Tradition was with


them; their ,nodel, college physics. had been featured by rigorousmathematieal discipline for a long time; problems requiring mathe-matical solutions are the easiest to devise and correct; to requirea class to solve several hundred problems was an assignment thatwas clear-cut and definite, and what is more, was easy to check forcomplete performance. Physics too frequently degenerated into ameaningless juggling of algebraic formulas, devoid of significancefor either physies or mathematies. A law would he studied andthen "proved"--as though it could heby means of a few read-Ligs taken in the laboratory f very likely judiciously manipulatedby the budding scientist la an attempt to make the "proof" moresatisfactory I.* Then would come the delugethe solution of largenumbers of problems based on the law and using the particularformula, which in so.ne mysterious fashion was a shorthand expres-sion for the laN,.

A reaction was inevitable and, as often happens with such phe-nomena, it went too far. It was advocated that physics should hestripped of its mathematics and made into a subjeot almost en.tirely descriptive. Thus Mieheison, one of America's foremostphysieists, proposes "for discussion the feasibility of a plan forthe teaching of physic's which avoids as far as possible the use ofmathematics of even the most elementary kind, and which givesto flu science of measurement only a secondary importance."Adams' suggests. "Mathematics should be used very little in the.class in physic's except for the solution of problems which are in-trodueed in connection with the laboratory work to generalize andestablish laws from given data."

Opinions such as these were widely circulated in the profes-sional literature of a few years ago. It seemed necessary thatevery high school physics teacher take a stand either for or againstthe demathematization of his subjeet. If it had not been for theinfluence of the colleges. it is possible that a form of physics zimi-lar to the old natural philosophy might have gained a foothold.As it is, the force of the movement is manifest in the perpetuationfrom year to year of various hybrid eourses in Household Physics,Applied Physic's. Physics for Non-College Students. and the like.Mot.- recently came a swing in the other direction. The cry went

Eumtpin 1,4 ru parted to havo said, "No amount of experimentation can ever!wave mP right. .1 single experiment may at any time prove me wrong."

4 Manisa. J. \V.. currrlatinn detwern Mathematfes and Phyitiest in American High.+r/o0/4. Master's thesis. Teachers College, Columbia Cniversity. 1902.


up that the basic outline for the physics course should be the samefor all. modifications in specific content being made to fit theneeds of individual classes. Above all, "demathematization muststop." a

As we look back over the contested issues from our vantagepoint in the year 1930, we realize that much of the time and energydevoted to the discussion of this problem has been wasted. Onceagain we have been the victims of our educational myopia whichrenders us unable "to see the forest for the trees." It is as thougha convention of carpenters should become engaged in a heatedcontroversy over the advisability of discarding the hammer infuture building construction, when no superior substitute is insight. The question is not whether they should, or should not,employ the hammer, but rather, how means can be devised toinstruct members in the more efficient use of all the tools of theirtrade.

So it is in physics. There is no question of whether we shallcurtail the use of mathematics as much as possible or expand itsuse in all directions. To debate the question is to distort and alterthe whole problem. Such discussion predicates a physics course,a main objective for which is: To show how the science of physicsmay he used to illustrate mathematical processes. Our problem isto devise means to employ more effectively this tool of the scien-tist's trade.

The Role of Mathematics in High School Physics. Rusksays in this connection:

Those who are crying for secondary school physics to throw off the burdenof mathematics and become descriptive. should carefully reconsider theirposition and what they mean by descriptive. Mathematics should certainlynot be loaded on the high school physics pupils as a burden, but withoutthe adequate use of mathematical forms neither methods nor appreciationof precise thinking about. physical phenomena can he developed. What isneeded to-day is not an attempt to develop embryo mathematical physicists.but a more frequent use of simple mathematical forms by all. Even inelementary physics the pupil should he Ird to look upon the mathematicshe uses as either simplifying the subject and making it more intelligible.or as making it directly applicable and useful. More mathematics than thepupil can thus consciously assimilate is useless and confusing.

Rusk is here suggesting certain specific aspects of the funda-11mulnli. D. P. and others, "The flare of the Numerical Problem in HighSehnol Phssieg." Xrhool Rye lor, Vol. 21i:39-13, 1;115.

"Rusk. lingers 5/.. Hour to Trash Phyairs. p 57. .T. B. Lippincott Co., 1923.


mental concept of mathematics as a tool. To these we can addcertain of our own, so that the list now stands as follows:

Mathematics can be used in connection with high school physics:1. To simplify the subject and make it more intelligible.2. To make it directly applicable and useful.3. To enrich Pie concepts of physics.4. To show the interrelations of the various divisions of the subject matter.(The various items in the preceding list are not mutually exclusive. A

single mathematical operation may be used in more than one of the aboveconnections.)

Illustrations of these uses will be brought out in a later sectionof this chapter.

But is this a conception of the role of mathematics in physicsthat is at all new? We have already seen how it was of use whenphysics was taugl t for its disciplinary values. Let us look atmathematics used in a different fashion by the mathematical re-search physicists h( fore we decide.

Many years ago, Nichols and Franklin in a preface to a textin physics said: "Calculus is the natural language of physics."Others have expanded the statement to include all mathematics.

A slightly different point of view is shown by the statement ofanother author, when he says, "The finished form of all science ismathematics."

With the recent discoveries in physics, the relationships betweenthe frontier physics and its language, mathematics, have subtlyaltered. Mathematics is often no longer the languageit is thespeaker itself. The scientific phenomena which mathen:atics hasinterpreted have been replaced by mathematical formulas whichare probably not capable of being translated into any sort ofmechanical modfsl. Indeed. we are warned against making theattempt.

This new variety of space. Einstein makes no attempt to visualize. Itsdefinition is strictly and severely mathematical. ... In such a space Einsteinhas found it po,sihle by means of the calculus of tensors; to build up a self-consistent geometry: and in terms of such a space he has formulated ageneral mathematical theory which as one special case reduces to Maxwell'sequations. and as another to the equations of Einstein's gravitational theory.'

I do not. maintain that this substitution of mathematical for-mulas for what we have been pleased to call physical reality is

* Hey], P. it., Neu. Front irrs of Phimics, p. 135. D Appleton & Co.. 1930.


universal in modern physics. I do maintain that it indicates achanged relationship.

The role of mathematics in secondary school physics, at thepresent tune, is probably midway between the two positions; the,me which it. occupied in early high school physics in which it wasused as a taskmaster to make the subject difficult, worthy of in-clusion on lists of subjects suitable for college entrance, and henceof great disciplinary value; the other which it occupies to-day inresearch physics. To our question, "Is this a new conception ofthe role of mat' ...Ales in physics?" we are forced to give a quali-fied negative. The Nile of mathematics as interpreter and simplifieris not new. It has simply taken on a greatly increased signifi-cance in high school physics of to-day.

An Integrated Physics Course. Before specifically illustrat-ing ways in which mathematics enriches high school physics, itwill be necessary to develop briefly the point of view of the lattersubject.

It has been evident for some time that one of the obstaclesstanding in the way of more satisfactory student. accomplishmentin high school physics was the manner in which the work wassegregated into five water-tight compartmentsMechanies, Heat,Sound, Light, and Electricity. Such a procedure made it difficultfor the student to grasp the underlying unity of the subject andhence to tie in each day's work with the course as a whole. Dis-satisfaction resulted, and poor learning was the usual outcome. Itseems necessary. then, to present the various divisions of physics.or any science, as part of a larger whole, or to state it differen+1.,,to develop the entire course around some large, unifying concept.This is not a new idea. nor is the concept difficult to obtain. Spacepermitting, it would he rather easy to show that the "Energy Con-cept" is the one best suited for such a development. Mann," asfar back as 1912, indicated how it. might be used. Numerouswriters of science works intended for popular consumption havetestified to its importance tHeyl. Bridgman, LuekieshTeans).In spite of this fact, textbook writers have lagged in producingtexts developed around the energy concept. A possible statemenl,of the concept for the physics course might be, "Physics is thestudy of energy and energy transformations which are basic to thecontinued existence of all life and to the universe itself."

Mann. C. R., op. cit.

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EXAMPLE, How much work can a steam engine, which is 8%efficient, perform for every pound of coal consumed?

The question obviously involves the relationship between heatand work. Is it a fixed one? The early work of Rumford andJoule. in particular, gives i;s the answer. Whenever a fixed amountof work is used up in producing heat, the quantity of heat energyproduced is always the same. The converse is equally true. Therelationship is that 1 B.T.I.7.*= 778 foot pounds. Each pound ofcoal when burned produces about 14,000 B.T.U. of which only8r,*i., or about 1,120 B.T.U., is utilized. Thus the engine can do1,120 X 778 or 871,360 foot. pounds of work. For each poundof coal, a 1,000 pound weight. could be raised about 900 feet!

b) Changing one form of mechanical energy into another formof mechanical energy.

EXAMPLE. An automobile, weighing 3,200 lbs., is travelling atthe rate of SO miles per hour. What is its kinetic energy?The problem in this form illustrates admirably what I mean

when I say that it is not a question of whether we shall or shallnot diminish the amount of mathematics to be used in high schoolphysics, but rather of the way in which we shall use it. At thispoint the student will have had the definition of energy as "abilityto do work" and of kinetic energy as "energy due to motion."And yet almost all texts are content to give problems which merelyrequire the student to work out values for kinetic energy, a pro-cedure which is almost sure to be meaningless for him. It is re-motely possible that by requiring the student to label his answerwith the proper unitfoot poundsthe unit of work, he %%ill obtaina fleeting impression that this kinetic energy may, in some man-ner, he converted into work. Let us make sure that he realizesthat this transformation does take place by adding the followingproblem:

The automobile hits a stone. wall and is brought to a stop.How great is the shock which the bumper receives, if it bends 6'"?

Using our ordinary formula, , , we learn that t2g hecar has a kinetic energy of 96,800 ft. lbs. This amount of workmust he absorbed in bringing the car to a halt.. Now work is ob-

The B.T.U. (British Thermal Vnit) is the an,ottnt of heat required to raisethe temperature of 1 lb. of W14 or t degree F.

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engine could cause the generator to develop 40 X 746 X 80% orabout 24,000 watts.

We may carry the question further if we desire by asking,"How many 40 watt lights could such a plant optratc?" The an-swer is obtained, of course, by simply dividing the 24,000 by 40,which gives 600.

d) Changing electrical energy into heat energy.

EXAMPLE. how long would it take to heat a liter (about onequart) of water from room temperature, 20°C., to the boiling pointin an electric percolator which uses 550 watts and which is 60701efficient?

The quantities involved are the unit of electrical energy, thewatt second or joule, and the unit of heat, the calorie, which arerelated by the expression. 1 joule 0.24 calorie. A calorie is theheat required to raise the temperature of 1 gin. of water 1°C.

The energy input into the percolator is in the form of wattseconds of which only 60% are available, due to unavoidable in-efficiencies of operation. The number of seconds is unknown. Letthis he represented by t. Then the number of watt seconds avail-able for transformation into heat energy is 60% of 550t or 330twatt seconds.

Now each watt second equals 0.24 calorie, so our 330t watt sec-onds is equivalent to about 79t calories. Putting it somewhatdifferently: in one second the percolator will produce 79 calories.

Let us turn our attention to the water. There are 1,000 grams(.1 liter of water weighs 1.000 grains) which are heated from roomtemperature, 20°C., to the boiling point, 1003C., a rise of 80 de-grees. Hence 1,000 X 80, or 80,000 calories, must, be supplied bythe percolator. Thus it will take 80,000/79 or about 1,013 sec-onds, which is about 17 minutes, to produce the desired result.

We could, of course, give other examples of mathematical illus-trations of energy changes, the conversion of light and other radiantforms of energy being the only ones beyond the scope of the highschool course.

A final energy change, most basic of t11, is that by which thesun and other stars produce their energy. ,Jeans and others haveadvanced as the most, tenable hypothesis that this energy is pro-duced by the direct change, or conversion. of matter into energy,mainly light and radiant heat. As a result of thi.: theory, we have


been deluged with such statements as these: There is enoughenergy in a piece of coal smaller than a pea "to take the Maure-tania across the Atlantic and back "; or, In a single pound of coal(or for that matter a pound of any substance) there is sufficientenergy to keep "the whole British nation going for a fortnight, do-mestic fires, factories, trains, power stations, ships and all." °

But is there a fixed relationship here, as with other energytransformations? Yes, for we arc told that for every gram of mat-ter completely destroyed 9 X lirF" ergs of energy are produced. Orif we change that to more familiar units, for every pound so de-stroyed about 3 X 101" foot pounds of energy result. Darrow '°reports that the earth receives (i0 tons of energy in the form of sun-light every year, or about of a pound a minute. That meansthat we are receiving in the form of light, radiant heat, and otherforms of energy % X 3 X10'" foot pounds of energy each min-ute. That is about 7.5 X 10'5 foot pounds per minute, which isequivalent to about 2.3 X 1011 horse power (230,000,000,000 H.P.).

All of these energy changes are made more real and vivid bythe realization that a certain quantity of one kind of energy maybe converted into another form and, by the proper mathematicalequation, the resultant quantity of new energy computed.

2. Definitions and Units. I have taken up the use of mathe-matics to illustrate energy changes first, simply bemuse thesetransformations are basic to all of high school physics. Thereare, however, other uses for mathematics than these. One of themost important is in clarifying and simplifying the defivitions andunits of the science.

a) Work is a unit often used. We define work as that whichis accomplished when a force acts through a distance, or, as it issometimes defined, "the overcoming of resistance." Soiretimesthe proviso is added that the force must be measured in the direc-tion in which the motion takes place. Not a very clear -rut defi-nition, possibly. But how simple it becomes when we put it in theform of a formula, W = F X s, and illustrate it by such a problem as this: -How much work du you do when you, walk to thetop of a stairway 10 ft. high.' Assuming you weight as 150 lbs.,the answer 1,500 ft. lbs. is obtained immediately.

Jeans, sir Juns. l'airursc .1 round l'x, p, 1S1, Tlie Macmillan Company,

1 Durru. L.. Thy Nrir 'World of Physical Discorf ry, 330. Iiubbx Merrill,1930.


We may use a sort of reductio ad absurdum method to furtherclarify the meaning by assigning and discussing such a problem as,"You are pushing a lawn mower which weighs 40 lbs. flow muchwork do you do in pushing it 10 ft.? Many will at first multiply theweight, 40 lbs., by the distance 10 ft., getting 400 ft. lbs. for ananswer. But going back to our fundamental definition of workas force times distance, we realize that the force required to pushthe mower is not given when we know the weight alone. Con-sequently the problem as stated cannot be worked. Let us carryour problem further. Suppose friction is negligibly small. Thenthe work done, the product of this negligibly small force and theten feet, is itself negligibly small. In the theoretical case whenthere was no friction at all, no work would be done. Suppose,though, we were to carry the mower to the top of a flight of stairsten feet high, could we then get an answer? Our formula requiresus to u::e a force. Do we know it? Yes, it is 40 lbs., because at theearth's surface it requires a 40 lb. force to lift a mass of 10 lbs.magnitude. Our work is now 40 X 10 or 400 ft. lbs.

By focusing our attention on the fact that we. must use u force,the formula has helped to eliminate the confusion resulting fromtwo situations in one of mhich we can use the weight of the ob-ject and in the other of which we cannot.

b) Another unit, the meaning of which is clarified by a ncithe-matical treatment, is the watt second, or the more convenient.larger, kilowatt hour. It is easy to establish the fact that the wattis a unit of power. Now, power is defined mathemat:eally aswork/time or Wit. A watt second then is a unit of power, 1171,multiplied by a unit of time, t. Dividing out the is we get simplyW. In other words, the watt second is a unit of work, or electricalenergy. Similarly the more practical kilowatt hour, which is merelya larger measure, is also a unit of electrical energy.

Other instances might be given but these are probably suffi-cient to illustrate this important function of mathematics in highschool physics.

3. Laws and Principles. One illustration will be sufficient,I believe, to show how mathematics helps to clarify and enrichsome of the laws of physics.

One of the most important laws in physics is the so-called"Reverse- Square Law" which states that the intensity of the illumi-nation from a given source varies directly with the strength of the


source and inversely with the square of the distance from it.Stated algebraically:

Illumination (foot. candles) = Candle powerDistance2

Let us compute the illumination at distances of 1, 2, and 3 ft.from a 36 candle power light. We get 36, 9, and 4 foot canc11.1s,respectively, numbers which bear the relationship of 9 to 4 to 1 toeach other or 32 to 22 to 12. The original distances were 1, 2, and3 feet. The illuminations are in the inverse order and squared.Certainly this more vividly illustrates the real significance of thelaw than a nonmathematical discussion could possibly do.

4. Structure of Matter. Without being inclined to demon-strate mathematically that it is all quite possible, populvr writersin this field have usually been content to astound with statementssuch as the following, the two parts of which are apparently con-tradictory.

If all the molecules in a cubic centimeter (1,4 of a cubic inch) of hydrogengas at ordinary temperature and pressure, were placed end to end in a singleline this "string of molecular beads" would extend several million miles ormany times the distance between the earth and the moon. In a cubiccentimeter of hydrogen at atmospheric pressure and at the temperature ofmelting ice there are 2.7 X 10' molecules, each having a diameter of 2.17 X10-1 cm.

(And then follows the part apparently irreconcilable with the for-mer statement about the "molecular beads.")

Less than a millionth of the total space [italics mine] is occupied by thehydrogen molecules under these conditions."

We may very well ask how it is possible that these moleculesshould reach so far when placed end to end, and yet that less thana millionth of the space should be occupied by them? Small won-der that high school physics pupils reading such a statement putit down as another one of those things beyond their comprehen-sion.

But let us see what mathematics shows. The distance themolecules will reach when placed in a line is the sum of theirdiameters. If we multiply the number of molecules, 2.7 X 10'9,by the diameter of each, 2.17 X 10-8 we get about 6 X 10" cm.This is 6 X 10" kilometers or about 3.6 X 10" miles, that is 3,600,000

111,acklesh. Matthew, Foundations uJ the Universe, p. 34. 1). Van NostrandCompany, Inc., 1925.


miles. Since the distance from the earth to the moon is only240,000 miles, we see that actually the molecules would reach 15times as far as the moon,

To check the statement about the space, we need to know thespace occupied by one molecule, that is, its volume. We will needto assume that the molecule is spherical in shape, which is prob-ably not quite true. The volume of a sphere is given by the formula:itad3, or % X 3.14 X t2,17 X 10 '')2, or about 5.1 X 10-2' cubiccentimeters. If we multiply this by the total number of moleculesin one cubic centimeter, 2.7 X 10'2, we approximately 14 Xor about 1/7,000 of a cubic centimeter. This does not agree withLuckiesh's statement. It may be he is in error. Or perhaps ourfundamental assumption that t:Ie molecule is spherical is wrong.If it were in the shape of a flat, elongated ellipsoid, the volumeof each molecule would be greatly reduced without affeeting thelength of the' molecular beads." Or we may attempt reconciliationby saying that th;s Ague. 1/7,000. represents the portion of thespace that Nvou 1 d he occupied by so/id spheres of the same diameteras the molecule, while it is known that the molecule is far fromsolid.

In any event, the fundamental picture of a volume of gas inwhich the molecules when placed end to end will reach vast dis-tanees, and yet in which the molecules are separated by distanceswhich are large in compatison with the size of individual molecules,is rendered quite consistent.

One further illustration of the use of mathematics in the fieldof the structure of matter is afforded by the kinetic-moleculartheory of gases. You will recall that this theory indicates thatth -s. molecules of gases are moving in a haphazard fashion in alldirections and that when two gases are at the saute temperature, themean kinetic energy of the molecules of one gas equals the meankinetic energy of the molecules of the second gas.

At some time during the work on heat. the fact is brought outthat the molecules of the gases are moving at different speeds andthat those of the lighter gases are the most rapid. Usually anexperiment styli as that We qrated is used to show this fact ex-perimentally. Whvti illuminating as is introduced under thebell jar, its molecules being smaller and of higher speeds diffusethroupli the porous cup faster than the umleettles of air inside es-cape to the (mashie. This produces an increased pressure inside the


cup which pushes the indicating column clown. On removing thejar, the reverse process takes place, a partial vacuum is producedinside the cup, and the indicating column rises to a point higherthan that at which it stood originally.

Now the explanation of this result, o.«4.44i.e., that the molecules of illuminating 130.20u.3 c.:t

gas +largely methane CH4, carbon mon-oxide CO, and hydrogen H) are alllighter than air and hence move at ahigher rate of speed, follows directlyfrom the theory which asserts that theaverage kinetic energy of the molecules 1"`"="6is the same. The mathematical state-ment is

1/4/1i2 WVa2 Dubruty2g 2g

where wi and vi stand for the averazi,\veight and average velocity cf the mole-oules of various substances in illuminating gas and w,, and r,, similarquantities for air molecules. Multiplying through by 2g we get

wivi2 = wova2.

Nov the average value for wi, the weight of the gas molecules, isless than for W. Hence in order that the equation may be pre-served we realize that the molecules of illuminating gas must havea haphazard motion decidedly faster than that of the air mole-cules.

If we desire to make the illustration more specific, let us takethe case of oxygen and hydrogen. Again we have

wovo2 = whvh2.

Nov it is known that the oxygen molecule is 16 times as heavy asthat 0/ hydrogen; in other words, that wo = 16wh. Substitutingthis value for wo in the preceding equation, we get

16whvo2 = whvh2 or 16vo2 V42.

Taking the square root of both sides4 vo = vh.

In other words, the average velocity of the hydrogen molecules isfour times that of the oxygen molecules.


This is the last of four types of illustration, chosen to bringout the various uses to which the tool, mathematics, can be putin high school physics. There are, of course, other illustrationsthat might have been used with equal validity. Let us reiterateour fundamental belief that no one is able to hand down ex-cathedra opinions on the quantity of mathematics to be used inphysics. We can only decide upon the quantity and nature of themathematics we will employ, in assisting in the development ofa certain specific unit, to perform those tasks of which mathe-matics is admittedly capable.

MATHEMATICAL ABILITIES NEEDED IN HIGH SCHOOL PHYSICSKilzer 12 has shown that, next to an interest in the subject, the

outstanding factor contributing to success in physics is ability tohandle the simple mathematics involved. This is merely a recentverification of something that many physics teachers have longbelieved. Most vigorous in their advocacy of this theory havebeen those teachers whom it helped to free from the responsibilityfor large numbers of student failures. When 25 per cent of thephysics class failed and this was not at all uncommon), theteachers could wash their hands clean of the whole affair by ex-claiming, "What can you expect with the poor mathematical prepa-ration our students get in this school?" Such an attitude is in-fectious and many very excellent physics teachers, at some timeor other in their careers, have rallied around the banner whoseslogan is, "Blame it all on the mathematics teacher."

As long as the doctrine of the transfer of training held sway,this attitude was bound to be widely accepted. To-day we realizethat., whatever may be the responsibility of the mathematics de-partment, the blame for the large number of failures that havebeen common in high school physics must be placed squarely uponthe shoulders of one individual, the physics teacher. With thisrealization the saner point of view, expressed in the followingquotation, has gained recognition.

The apparent lack of transfer of training is due to both the failure toretain what was learned and the failure to m, any common connectingelements between the field of mathematics and mathematics in physics. Poorwork in the mathematics involved in physics is not entirely due to the

" ilzvr, L. R., ."rhe nitheolattes Needed In High School Physics," RehoolKoiener and if at hem ( Mem , Vol. 211 811).31;::, ( An abstract of the author'sPh.D. dissertation on this subject.)


mathematics itself. . . . Thus mathematics and the understanding of thesubject are dependent on ability to read and understand material connectedwith physics; to understand what is said by the teacher and others in theclass."

Before attempting to solve the problem of what to do about theacknowledged inability of students to solve numerical problems inphysics, two questions need to be answered:

1. What mathematics is actually needed for the working outof the problems in physics?

2. How well can the average student handle this mathe-matics?

Reagan 14 solved the 241 problems found in one edition of Milli-kan and Gale's text, and analyzed each for the skills needed. Hefound that arithmetical processes were the most commonly used;addition being used 47 times; subtraction 37 times; multiplication422 times; division '266 times; common fractions (including allprocessesreduction, addition, etc.) 97 times; as well as miscellane-ous skills with smaller frequencies.

Algebra was used less often, the skills with their frequenciesbeing:

1. Trarvlation of laws of physics into mathematical formula 9

2. Derivation of formula from given mathematical relationship 63. Selection of formula 56

4. Solution of equation with one unknown 11

5. Solution of quadratic equation(Not solvable by factoring.)

6. Squaring a binomial7. Operations with signrd numbers 13

Geometry was utilized with still less frequency. Only 16theorems, of which the most important were those dealing with thesimilarity of triangles. proportionalities between lines, and the re-lation between the sides of a right triangle, were found.

The use of trigonometry, solid geometry, and the like, was notclearly indicated as necessary.

As a result of this study, Reagan concludes that the demandson mathematical ability arc not unduly heavy; that the knowl-edge of arithmetic is satisfactory if the student can multiply and

13 (in s. Mildred J.. auxex of Failure in High Nehool Physire. Contributions todunthm, V.I. U. World Book emnintny. 1n24.

"Reagan. ;. W.. "The Mnthennit lex Involved In Solving High School PhysicsProblems.- ,Shwil Nvit /UT and Mathcmatirg, Vol. 25 : 292.299, 1925.


divide integers up to 12 places and if he can app the laws ofmensuration; that the ordinary geometry course is reasonablysure to be adequate preparation; but that algebra is the mostlikely to be deficient where work with formulas, ratios, and pro-portions is involved.

Lohr 15 prepared a 28-problem test which he administered toclasses in both secondary and junior college physics, He founda rather large list of specific mathematio4kabilities, such as theability to determine the volume of a sphere given the radius, tomake a line graph of the relation 1.1F. 321 and the like,which could be solved correctly by less than 75 per cent of themembers of both groups; and a somewhat smaller list, such as thesubtraction of decimals, and redwing inches to centimeters, thereduction factor being given, that could be handled correctly bymore than 75 per cent of the college group, The secondary groupwas not significantly different.

He concludes that, on the whole, "pupils come to physics witha marked ability to handle the mathematics of physics"; that theinabilities can be determined; and that it is the duty of the physicsteacher to identify the mathematics difficulties, to reteaeh themathematics needed, and to teach the physics of the problem situa-tion. Note the temperateness of this last conclusion. There is notendency to lay the blame on the mathematics teacher. Physicsteachers have come to realize that be a student as well trained asyou please in mathematics, unless the physics in the problem situa-tion is well taught there will be an unsatisfactory accomplishment.The difficulties inherent in the situation, even when all these con-ditions have been met. is well brought out by Nyberg's 1' criticismof Lolly's work. The writer points omit that some of the inabilitiesrevealed might he (Inc to small differences between time wordingcommonly employed in physics texts and that used in algebras.These small differences are very likely not to he explained away inteaching the physics of the problem. For example. one of the skillsin which large numbers of the students were deficient was theability to determine the per cent of error between the true and timemeasured scores. Nyberg points out that a stident might fail to

15 I.nbr. "A Study of thy Ma them:Wm! AM lIti Powrrs awl Skillsn s Shown by Certn In Classes In PhysIvn I Svirnep. Frhunl Nriener and Mathe-motirA, Vol. '271! k!:-1 R14. l 92n.11 Nyhtig. .1 Hs. I II:4E11.44ln 1.f an .1 rt on :Mu t Iwnin t Iva I .1bIlitIPs andsrhool Srien re and Mat honntirx, 211 : 9.15. 19'211.


attempt the problem through not knowing whether to compute theper cent error on the true or the measured score, even although,when the scores are reasonably close, results are virtually the santewhichever is employed. Algebra texts commonly indicate which touse as a base.

Similarly, graphing the relationship C.= (F. 32) % mighthave seemed more difficult than it is intrinsically, because the unitsfor graphing were not stipulated. In other problems, Nybergseemed to find violations of graphing rules, hazy instructions, awlunfamiliar terminology creating artificial difficulties.

All this serves to illustrate the complexity of the whole prob-lem. Beside teaching the physics of the problem situation, theteacher, in order to secure reasonably satisfactory performance inproblem solving. will probably need to reteach the mathematicsinvolved, identifying identical elements, and should also make surethat unfamiliar phraseology or deviation from mathematical prac-tice do not needlessly complicate matters. It is quite probable thatKilzer's suggestion that a pretc t of mathematical ability he givenearly in the course is a good one.

WHAT PHYSICS ASKS OF MATHEMATICS

All of the suggestions of the preceding paragraph were basrdupon the assumption that the mathematical training of enteringstudents is satisfactory. Let us consider what steps may be takento make this preparation of greater value in plip:cs. takinpReagan' s IT analysis as a starting point for our progra mathe-matical training.

1. Algebra. Arithmetic is not commonly taught in high school.so first we will consider certain specific suggestions with respect toalgebra.

a) Formulas. An examination of recent textbooks in algebrawill reveal that a great deal more work with the formula is indi-cated than was common in earlier books. It is very probable thatthe type of wo k being done in this line in the best of modernschools is amply odequate for the needs of high school physics.

More specifically, a physics teacher might suggest that ample

practice he given in solving a formula such as I = -F fc,r both E

and R and, in general, that drill he given in solving each of theIT Ittqlg:1 II. 1i. NV


formulas commonly encountered in physics for letters of the right-hand member, In addition, considerable practice should be givenin obtaining the value of one symbol when actual values for theothers are given.

t4onietinies physieS students are asked to examine data obtainedin the laboratory or elsewhere, in an attempt to discern the rela-

tionship connecting two of the quantitiesinvolved. Sometimes the data are exceed-ingly simple and the relationship quiteexact, as in the table at the left:

Physics teachers will recognize these asdata from the experiment showing the re-lationship between resistance of various

portions of a circuit and the voltage drop around that portion. Analaebra student will easily discern that there is a direct relationshipbetween the two quantities. True, it is not quite exact, but in viewat the fact that these are experimental figures and hence subjectto the usual errors of experiment, a consistent mathematical rela-tionship is definitely suggested.

Data do not always come out so nicely: as, for example. in thetable at the right which gives inea:-urcd values for the distancecovered by a -frictionless" car, movingdown a taut wire, in various time intervals. Pis-TA :leg

I.r.Ntvro or Vol.TA0k1S I I" 'IL M WI Its 111101'

23 cm.50 cm.75 ctn.

100 cm.

1.5 volts3.1 volts4.5 volts5.9 volts

Here the relationship is at once lessobvious, and less exact. If thy.! -tudent isto notice at all the tendency tor a directpl'oport ionality to exist between the dis-

14" 1 sec.4' 5" or :53" 2 sec.10' or 120" 3 sec.17'2" or 206" 4 sec.

tance and the square of the time, he needsconsiderable experience with a similar type of work.

It is quite common in some high school algebra classes to askstudents to write a formula that will express the relationship be-tween the quantities in a table. The work of the two tables abovesuggests the desirability of including: a certain number of problemsill which. as the data are supposed to he the result of actual meas-urement (and hence subject to error) the simple mathematicalrelationship between the quantities may be slightly in error forsome, or possibly all, of the pairs of measurements.

b) Proportion and Variation. Physics texts abound in suchexpressions as. "The density of air. or any gas, varies directly asthe pressure at constant temperature." And then as a mathemati-


tcal equivalent of this law sonic such expression as

Dt P= is given.I

The first time this situation arises (and it arises often), the begin-ning student is plunged into difficulty. In many instances he hashad no experience in dealing with qwntities that vary directlyone with the other (although he may 1. se handled an equivalentproblem disguised under a different terminology), or if he has, amethod not employing a proportion may have been used. Themathematics teacher will be of very real assistance here if the use ofthe subscript. is taught. to bring out the distinction between differentquantities of the same kind. (Thus D. D,, D, or Da, Di D, mayindicate different numerical values of density.) In addition, the

Di Piuse of expressions such asD /I,:--

.,= to show a direct variation be-

.tween two quantities, and Pt = vt; to indicate an inverse variation,

should be thoroughly taught and utilized in a number of situations.D P D1 D. DThe fact that and the like, are all

P2' P, Pr'equally valid as expressions of the same direct relationship, as wellas the corresponding equivalents of the inverse variation, such asthat given above, should also be brought out and thoroughly fixedby drill.

The method for solving proport'ons should also be taught. Inaddition to the usual cross-product (product of the means equalsproduct of the extremes) method, the various ways for first simpli-fying a proportion should be made functional. The followingproportions will make this point elem.. In each case before cross-multiplying the indicated simplification should be performed.

9= 9 Simplify dip first, ratio to r.

257

Divide both nm 2uerators by 5, chan. ng to the formx

10

Divide both denominators by 4 (equivalent to multiplying45 x both :Ws of the equation by 4), changing to the form8 12 45 x

2 3'c) Other Operation$, The other algebraic skills specifically

indicated by the stmlies of Reagan and others are likely to bethoroughly mastered in the usual algebra cour,o, since while they


occur with a small frequency in physics problems they occupy amuch more prominent place in the mathematics course. For in-stance, Reagan found that in the 241 problems in the Millikan andCale text only I solution of a quadratic equation and only 13operations with signed numbers were specifically required.

While not fundamental to the solution of physics problems thereare several topics, not commonly included in an algebra course,which bear on the ability to handle the mathematics of physics.Taught in the algebra class, they would at once lighten the physicsteacher's burden and add topics which would be of interest inalgebra.

d) Metric System, Even though many algebra students maynot go on to take a physics course, yet the increasing use of themetric system in everyday life amply justifies its inclusion in themathematics course. The increasing tendency for authors ofalgebra texts to include a short section on the metric system, bearstestimony to the value of such a unit and to its fitness in a modern,laboratory-type course. An opportunity is here afforded for cor-relation betty een the moheniatics and the physics departments.Metric rules, grain weights and balances, graduated cylinders andliter measures, all can be borrowed. Where possible, the meetingplace of the algebra class may be changed to the physics laboratoryfor the duration of this unit. The first strip-film roll from the seton Mechanics (sold by the spencer Lens Co.), which deals witht'x metric system, may be shown with advantage. Not only issuch cor7elated work helpful foi the physics teacher in renderingunnenessary any detailed study for the metric system, but it is aninteresting variation from the usual routine of the algebra class.

e) The Slide Rule. If we accepts the theory of the ride ofmathematics put forth earlier in this article. we are forced to theconclusion that any device which simplifies the mere mechanics ofmultiplsing, dividing, and the like, is worth while. The slide ruleis such a device. We may summarize its advantages as follows:

1. By reducing the amount of attention that nerds to be devotedto tl,e arithmetical processes by which the answer is obtained,it allows the focusing of a more undivided attention upon thephysical principle involved.

2. When its use is thorow_dd mastered, a great deal of time ;ssaved. A greater number of problems may be worked, .a'


the time spent on the mathematical development of a giventopic reduced.

3. Its use prevents obtaining results with an entire* fictitiousaccuracy. If the measurement of the diameter of a eirel,gives 2.46 cm., the final answer cannot be accurate in agreater number of places. We are all too familiar withanswers such as 4.75292664, which is obtained with the usualarea formula, multiplied out "longhand" using n = 3.1416.In this answer, the latter decimals are of no significance what-soever as they were all obtained by a multiplication involvingthe doubtful measured figure 6 of the original measurement.The greatest number of places to which we are justified incarrying the answer is three, giving 4.75 sq. ctn. for the area.The slide rule more or less automatically takes care of suchsituations.

Not only is the slide rule of advantage in physics but the mas-tery of its operation is quite within the grasp of a ninth gradealgebra student. In fact the manner in which multiplication anddivision are performed can he taught in a very few minutes.

The scales of a slide rule are logarithmic. Because of the de-creasing space between successive numbers of the same order, thespacing in different parts of the rule is not the same. Between 1and 2, each of the smallest divisions indicates a value of .01 of thewhole space; between 2 and 3. and 3 and 4, .02; and from 4 to 10,the end of the rule. .05 of a \dole space. This indicates the follyof stopping with a mere demonstration of the principle of the rule.We must insure an ability to interpret the scales quickly andaccuratelyan ability which can be acquired only after consider-able drill. The algebra student who, by .i.; inself. will acquire anyreal ability to use the rule following a short demonstration is anexceedingly rare individual.

The Keuffel and Esser Company, 27 Fulton Strect, New YorkCity, who have rules that retail for as little as 75 ceeits nr onedollar (subject to a school discount). also manufacture a large 8'demonstration rule, and furnish helpful suggestions on methods forteaching the use of the rule. In my own work. I have foundmimeographed work sheets helpful in this connection. On one suchsheet, three drawings of the 10" scale. full size. were shown. Oneshowed only the major divisions. the second both major and sec-


ondary, and the third drawing some of the smallest divisions.Questions were asked about the size of the different divisions andstudents were required to indicate the value of certain points indi-cated by arrows on the drawings. On a second sheet, magnifica-tions of portions of each of the three size divisions were drawn.Arrows were directed to various points on the drawings. By esti-mating the reading of each arrow, practice in using the scales, whenthe indicator does not coincide exactly with one of the markings,was afforded the student. Other work sheets give instructions forperforming the various operationsmultiplying, divid!ng, squaring,and extracting square root, together with a selection of problemsand answers.

To do a thorough job of teaching the slide rule is not the easiestof tasks, but it gives the student a tool which is useful in manyfuture activities. Furthermore, students like the work. In the firstflush of their enthusiasm they will not multiply 2 X 3 except onthe slide rule. It is the task of the algebra teacher to utilize thisearly enthusiasm to secure for the students a degree of confidence inthe accuracy of their results which will insure their turning to therule as a convenient tool, and not as a novelty or toy.

Since the slide rule is based upon logarithms, many teachersmay feel that. a knowledge of these is necessary for mastery of theinstrument. This does not follow. In industry many persons usethe rule who have only the vaguest of notions of the principlesinvolved. That these persons are any the less accurate or rapid intheir use of the rule has yet to be shown.

Criticisms of the slide rule sometimes advanced are that it isinaccurate and that it encourages careless work. The slide rulehas its limitations. of course, but it is amply accurate for mostcomputations whieli high school students are called upon to per-form, particularly where measured quantities are involved, as inphysics. By bringing out the principle that an answer obtainedby using measured quantities cannot be more accurate. except inthe case of an average, than the least accurate quantity employedin the computation. the value of the slide rule in automatically-rounding-off" n11:.:Wcr:-.: Will he indicated and fictitiously accurateanswers will be eliminated. A discussion of significant figures canbe appropriately taken up in this connection. To the charge thatit. encourages careless work, I peed simply respond that one of tin'most important uses to which the slide rule is put, is that of check-


ing results of ordinary computation. In the long run, the use ofthe slide rule, by facilitating the checking of results, will probablyincrease accuracy. Any individual cases in which carelessness hasfollowed its use are likely to be the result of failure to impress thelimitations of the instrument upon its user.

f) Exponents. Any reader of popular science articles cannothelp but be impressed by the number of numerical quantities suchas 2.7 X 1019 or 2.17 X 10-8 which occur. This would suggest thedesirability of adding to the usual work on exponents a more de-tailed consideration than is commonly accorded this portion of thesubject. This work should include raising to a power; extract-ing a root; multiplying; dividing; changing a decimal to this form;changing from this form to a decimal. If, in addition, a real feelingfor the bigness of such a number as 2.7 X 1019 and for the minute-ness of such a quantity as 2.17 X 10-8 is developed, at valuableservice will have been rendered the student.

2. Other High School Mathematics. The investigations ofReagan and others have revealed that the small amount of geom-etry (16 propositions} required in high school physics is almostcertain to be sufficiently functional after the usual "exposure" tothe subject.

Since physics is taught somewhat more commonly in theeleventh sehool year, algebra and geometry constitute the usualpreparation of the pupils in mathematics. This being the case, aknowledge of trigonometry cannot be ,xpected. Knowledge of thesimple functions, sine, cosine, and tangent, and also an ability touse a table of their natural values are helpful, but not necessary.This small amount of trigonometry is often learned in ninth yearalgebra courses.

Some advanced schools :ire offering courses in the elementarycalculus in the senior year. With a physics group composed ofseniors taking both subjects, there is an opportunity for correlatingthe work Of the two subjects in a few instances, notably in acceler-ated motion. Thus, taking the derivative of the eqeation for thedistance covered by a freely falling holy in a given time, S=1/2gt2,we get dS/dt = at, But dS/dt is simply a definition of velocity,so we have = gt. Similarly, tai' ng the derivative of this equa-tion, we get de/dt = g. Again, (Iv/dt is a definition for accelera-tion. These processes will give a better insight into the meaningof the terms -acceleration" and "velocity."


Tyler I° puts it (speaking of a college group), "I like to tell mystudents that the court will not accept v = sfi as a defense in acase of over-speeding; that the automobile trap for obvious reasonsIlsesA

' and that our definition of derivative as speed merely coin-Atpletes the transition to the limit."

Of course, there &re no problems in the ordinary physics textthat require a knowledge of calculus for their solution.

CORRELATED MATHEMATICS AND PHYSICS

There have been widely varying opinions expressed as to theamount of physics that should be introduced into a mathematicscourse. Young says that physics "should be taught simultaneouslywith mathematics throughout the four years of the course, bringingthe mathematical theory and the physical application into closejuxtaposition." '" In England there is customarily a great deal ofcorrelation between the courses. Nunn, an English writer, however,atlirins that problems involving science principles to he used in themathematics course, "must be limited to those whose solution issimply the question of the straightforward mathematics." 2" Ihave, I hope, made my own position clear with regard to that whichI consider to be the role of mathematics in physics. As to therule of science in the mathematics class, that is beyond the scopeof this article. Suffice it to say that I am perfectly willing for themathematics courses to "help themselves" to as much science asthey see fit in order to enrich the content of their subjects.

And yet, in that which I am about to say, I may seem to beinconsistent. There have been at various times enthusiasts whohave insisted that a combined physics and mathematics coursemight be worth while. In at least one case, this cook the form ofcohcbining three subjectsa year of plane geometry, the mathe-matics of the eleventh school year, and a year of physic's into asingle course to run for two years. The total number of recitationsin the two years were equal to those of two "majors." It wouldbe most unscientific for me to attempt to decide finally upon the

1"1 Tyler. II. W., "NlatlzmuatIvs In $e1,w ." The .Itathctttaeirx Tcar,her , V(11.21:273-279, 192s.youn. 3. w. A., Tar Tr,trhin, of mat h, mat irx. Longmans. (irpen and (Nan-puny. 192:i.

0, Nunn. T. ppro,%. hc fi of 1 fp Va, I.ungtuan. 1;rovn antl Company,1919.


merits or demerits of such a plan. It can unquestionably be carriedout with a good group of students who will remain through thetwo years; and thae will be saved. In the same way, I am surethat a gnaw of equal ability could he prepared to take the CollegeEntrance Examination in physics in one-half to two-thirds theusual time. Unquestionably it would be a course in which manyof the larger generalizations and principles of physics would beslighted, in the hurry to obtain the necessary information andabilities (largely mathematical) necessary to make this scholastichurdle. In the same way, I feel that many of the larger principlesof physics would be lost in sonic. of the forced correlations of thiscombined mathematics and physics course. Instead of ny realappreciation of the energy concept and what it means to man andhis universe, the student would probably gain a very real under-standing of the fact that science makes excellent illustrativematerial for the study of mathematics.

SUMMARY

In this chapter. I have attempted to trace briefly the develop-ment of to-day's high school physics, to show how it changed froma highly descriptive, nonmathematical subject to one N1 Moll, underthe domination of the colleges, and in virtue of the doctrine ofdiscipline, became a highly mathematized subject, and how finally.during the last decade, the pendulum has swung back, going per-haps too far in its swing.

Whatever may be the final solution to this problem, it can bedemonstrated, I believe, that the role of mathematics in high schoolphysics is somewhat different from that which it assumes in thecolleges. and decidedly different from its dominant position onthe frontiers of physics research. By illustration, the attempt wasWade to bring out the nature of this role. i.e., that mathematics isan invaluable tool for simplifying, clarifying, and enriching variousaspects of the subject.

A consideration of the amount of mathematics training neededfor this function followed. and the conclusion was reached thatwhile the average student in physics has been "exposed'. to enoughmathematics. for various reasons, he is unable to work manyquite simple problems. The fault cannot he laid at the door ofany one department of the school, being inextricably tied up withthe newness of the subject for the student, and with the smallness


of transfer. With the idea of improving, if possible, the mathe-matical preparation, suggestions for slight changes from and addi-tions to the usual algebra course were given.

If, as the result of this chapter, any clearer understanding ofthe character and scope of the problem and of the relationshipbetween mathematics and physics is attained, the purpose withwhich its writing was undertaken will have been realized.

POLYGONAL FORMSB GE()1U 1). BIRKIIOFF

Harvard l'aiersity, Cambridge, Mass.

The empirical aesthetic formula

M = 0/Cwhere 0 is the "order" attributed to certain aesthetic factors (suchas symmetry, etc. ) with proper weights attached, where C is the"complexity," and .1/ is the "aesthetic measure" itself, finds itssimplest possible application in the interesting, if elementary,aesthetic question of polygonal form. Our aim in this chapter is todeal with this case, and exhibit the result of the rigid application ofthe formula to ninety typical polygonal forms, ordered accordingto deereasing values of 41/ (pp. 190-195), If the reader finds agradual diminution in attractiveness in passing from the firstpolygon (No. 1) to the last (No. 90), the formula may be regardedas substantiated.

The judgment of students in two graduate courses, held atColumbia University (summer 1929) and Harvard (summer 19301,seems to indicate the validity of the formula. I wish to expresshere appreciation of the cordial eoliperation which I received fromthe students in these classes. In one instance. that of the right tri-angle resting on a side (No, 70) , the rating was felt. to he toolow. If, however, the context supposed in this connection, namely,that a sinfile polygon is used as a tilt in vertical position, is keptin mind, it will be clear that while the right triargle is valuedhighly as an element in composition. it is scarcely ever used inthis particular way, i.e.. in isolation.

Iii judging the validity of the formula it is necessary of courseto eliminate all accidental connotations, such !' .4 the religious oneof the crosses in the list, that of a rectangular box suggested by

i'rf,ert dinqv tt. the int, ettut,,,nra Math( matff i rimy/INN, ItoIngtm. 1928. Forn 4.1 II" i'''."1)"1"gioni bait: Of OW (M.11111111, SIP an art bit. -.1 Math°.mat al .1i.i.n.a4.11 lu A.Si Ill°1 to appear in Nientift. I xpeet topub1i:-11 %anon:4 applia htwk form as swat as possibit.. - non.

165


No. 37, and the like. Furthermore, the first effect of novelty mustbe discounted.

1. Preliminary Requirements. If the problem of classificationof polygonal forms is to have reasonably precise meaning, someparticular representation must be chosen. We shall accordinglyimagine that we have before us a collection of blue porcelain tiles.of uniform material and size. It is easy to consider these solely intheir aspect of pure form.

A further requirement must be imposed in order to fix the psy-chological state of the "normal observer." Such a polygonal tileproduces a somewhat different impression when it is seen upon atable than when it is seen against a vertical wall. In fact, such atile lying upon the table would be viewed from various angles,while on the vertical wall it would have a single favorable orienta-tion. Therefore it is desirable to think of the polygonal tiles assituated in vertical position against a wall. In general, the selectedorientation will of course be the most favorable, although it neednot be so, as for instance in No. 85 o:* the list.

Perhaps tile actual use of the polygon for decorative purposeswhich most nearly conforms to these conditions is that in whichsome selected porcelain tile is set at regular intervals along astuccoed wall.

Just as in all other aesthetic fields, a certain degree of fan aw-ay with the various types of objects involved is required beforethe aesthetic judgment becomes consistent and certain. The ninetypolygons listed in order of decreasing aesthetic measure will furnisha fair idea of the extent and variety of pol.gonal forms.

It is clear that when these requirements are satisfied the prob-lem of polygonal form becomes a legitimate one.

2. Triangular Form. With these preliminaries disposed of,let us turn to a determination of the principal types of aestheticfactors affecting our enjoyment of polygonal form. Once such adetermination has been made, we will be prepared to assess therelative importance of these factors, and to formulate an appro-priate aesthetic measure such as we are seeking. We will beginwith the simplest class of polygons. namely the triangles.

Now triangles are usually classified as being either isosceles andso having at least two sides equal, or scalene. Clearly the isoscelestriangles are more interesting from the point of view of aestheticform. The proper orientation of an isosceles triangle is naturally

POLYGONAL FORMS 167

one in which two equal sides are inclined at the same angleto the vertical. This is the case for each of the triangles (a), (b),(0 of the adjoining figure. In all of these, the triangle "rests"

(a) (b) (e)

FIGURE 1

(d) (c)

upon a horizontal side. However, if these triangles are inverted.the equal sides will again be inclined at the same angle to thevertical. It is readily verified that this second reversed orienta-tion is also satisfactory, and that these two are superior to allothers. With these orientations only do we obtain "symmetryabout a vertical axis." This is clearly a desideratum of first im-portance.

Obviously if a symmetrical figure be rotated about the axis 1of symmetry through 180', it returns to its initial position.

Judgments Of symmetry about a vertical axis are constantlybeing made in our everyday experience. Let us recall, for example,how quickly we become aware of any slight asymmetry in thehuman face. Thus the association "symmetry about a verticalaxis" is intuitive, and is pleasing to a notable degree in almostevery instance,

If the isosceles triangle (b) be made to rest upon one of thetwo equal sides, there still remains the feeling that the triangle isin equilibrium, although the symmetry about the vertical axis isthereby destroyed. It will be observed, furthermore, that the sym-metry about the inclined axis is scarcely noted by the eye and isnot felt favorably. Thus the triangle in its new orientation makesmuch the same impression as any scalene triangle which rests upona horizontal side compare with ( I. The indifference of the eyeto such an inclined axis of symmetry is also evidenced by theisosceles right triangle (d) with one of its equal sides horizontal.

On the other hand, if the isosceles triangle (b) he given anyorientation whatsoever other than the two with vertical symmetryand the third just considered, there is dissatisfaction because ofthe lack of equilibrium.

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MOOEIIIVHA H.LXIS HILL

POLYGONAL FORMS 169

grouped in the following five given classes in descending order ofaesthetic value: (11 an equilateral triangle with vertical axis ofsymmetry; 12) an isosceles triangle with vertical axis of symmetry;(3) a right triangle with vertical and horizontal side; (4) a tri-angle which is without vertical axis of symmetry and rests upona sufficiently long horizontal side to insure the feeling of equilib-rium; 15) any triangle which lacks equilibrium. The trianglesof the first three classes are definitely pleasing; those of the fourthclass are perhaps to he considered indifferent in quality; and thoseof the fifth class arc definitely displeasing. Since it is a naturalrequirement. that the best orientation of any triangle he selected,the fourth class will contain all tl... scalene triangles without aright, angle. and the fifth class will not enter into consideration.

It has been tacitly assumed in the above analysis of triangular'form that no side of the triangle is extremely small in comp'1....onto the two other sides, and that no angle is very small or very nearto 180'. These arc obvious prerequisites if the triangle is to becharacteristic. If they are not met, the triangle approximates inform to a straight line and the effect is definitely disagreeable,because of ambiguity.

We are now in a position to list the aesthetic factors that havebeen thus far encountered: vertical symmetry (+), inclined sym-metry (0), equilibrium (+I, rotational symmetry (+), perpen-dicular sides (0). diversity of directions (), small sides (),small angles or angles nearly 180' ().

Here and later we use the symbol f+) to indicate that thecorresponding association or element of order operates to increaseaesthetic value, the symbol 101 to indicate that it is without sub-stantial effect, and the symbol to indicate that it diminishesaesthetic value. Such associations or elements of order will accord-ingly have a positive index in 0 if the symbol is (-F.), and a nega-tive index if the syndic)! is t 1.

3. Plato's Favorite Triangle, It cannot he emphasized toomuch that the classification of the various forms of triangle givenabove takes into account only the simplest and most naturalaesthetic. factors. I low completely such a scheme of classificationcan be upset by the introduction of factors based upon fortuitousassociations, is easily illustrated.

Plato in the Timacus says: "Now, the one which we maintainto be the most beautiful of all the many triangles and we need


not speak of others) is that of which the double forms a third tri-angle which is equilateral." The context makes perfectly clear inwhat sense this statement is to be interpreted: If one judges thebeauty of a triangle by its power to furnish other interestinggeometrical figures by combination, there is no other triangle com-parable with this favorite triangle of Plato. For out of it can bebuilt (Figure 2) the equilateral triangle, the rectangle, the paral-

FIGURE 2

lelogram. the diamond, and the regular hexagon among polygons.as well as fi'r'e of the five regular solids. This power in combina-tion was peculiarly significant to Plato, who valued it for purposesof cosmological speculation. It was on such a mystical view thathe based his aesthetic preference.

Yet it may well be doubted whether persons not having hisparticular philosophic outlook would agree with Plato. In fact,it appears that this scalene right triangle is not superior to thegeneral right triangle for the aesthetic problem under consid-eration.

4. The Scalene Triangle in Japanese Art. It is well knownthat the Japanese prefer to use asymmetric form rather than thetoo purely symmetric. Indeed, in all art, whether Eastern orWestern, obvious symmetry tends to heroine tiresome.

In particular it has been said that all Japanese composition isbased upon the scalene triangle. Is this fact in agreement with theclassification effected above which concedes aesthetic superiority tothe isosceles and in particular to the equilateral triangle? Theanswer seems to be plain: When used as an element of compositionin paintino. the isosceles triangle may introduce an adventitiouselement 01 symmetry which is disturbing to the general motif. But.

he much more elementary question of triangular form per sr,the general opinion. at. least in the West, is in favor of the equi-lateral and isosceles triangle rather than the scalene triangle.

Recently while in Japan I was fortunate enough to be able to

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tLt SK11041 IFXODAPIOtT

172 TII E SIX TII YEARBOOK

least angle of rotation is eontained an odd number of times in360° will not posess central symmetry; for example, the equi-lateral triangle is a case in point, with a least angle of rotation120° which is contained 3 times in 360",

It has been claimed that the rectangle is a form superior tothe scuare and even that certain reetangles such as the "GohlenRectangle" excel all others. I shall indicate later in what- sense,if any, sueli statements eau be valid (Section 6).

In the second ease of symmetry about the vertical axis, thequadrilateral has two of its vertices on the axis of symmetry, butnone of the sides intersects the axis. Here the general possibilityis indieated by tel and t(1) of the figure below in which quadri-

(aaI (hi

4

i I (di

lateral lc) is roex ;old (it) is reilntrant. The first of these may,however, reduce to Cie. equilateral quadrilateral or diamond asin an, or even to the square (a) with sides inclined at 43° to thehorizontal directio.,.

Of the two g.T11("%ill represented hr the quadrilaterals ir)and id} in Figure 4, . lc ar that die eonvex type () is definitelysuperior to the altetto,:h.e -ei:nt rant tne of (oadrilateral (d). Thisreilntrant character evidently operates so that the quadrilateralsug-g('sts a triangle from Nellie]) a triangwar -niche- has been re-MoVI I11 general cavil polygon ly UPlv4.Tn a given reintrantpolygon and t (011Vn pillyu;v11 %%111(11 (11(10S('S it ktermed a -niclu of the rei:ntrant polygtol. A rubber band stretchedaround the given polygon will take tile form of the minimum en-closing convex polygon.

It is not the men, fact that the quadrilateral is rei'utrant whichis (leciiely unfavorable. Consider, for example, the hexagt..nor six-pointed star (see No. ti, page 160). This star is evidently

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follows the trapezoid, with two sides parallel as in (b) ; evidentlythe association "parallel" is one made constantly and intuitively ineveryday experience. Then follow the general convex quadri-lateral like (e), and finally the reentrant case illustrated by (d).

/(a) (h) le) (4)

notrain 5

It will he observed that the presence of an isolated right angleor of two equal sides, as in Irf , is without special influence.

We have now examined the various types of quadrilaterals andarranged those of earl, type in order of preference. It remains tocompare briefly those of different types. On the basis of my ownaesthetic judgment I am led to arrange them in the following orderof diminishing aesthetic value: the square; the rectangle; the dia-mond; the convex quadrilateral symmetric about an axis throughtwo opposite vertices, the symmetric trapezoid and the parallelo-gram; the reentrant quadrilateral symmetric. about an axis; theconvex quadrilateral without symmetry; the rei:ntrant quadrilat-eral without symmetry. Of these, the last two types are fletinitclyunsatisfactory. Here I assume that the quadrilaterals are placedin the most favorable position. of course. This iVe arrange-ment is that assigned by the aesthetic formula.

We shall not attempt at this stage to compare triangles andquadrilaterals with one another.

There are two new types of elements of order brought to light1w our examination of quadrilaterals. The first is of negative type

-1 and oiler:Iles when the quadrilateral is reilntrant. Furtherinsight into the nature of this element will he obtained in a follow-ing section. The second is connerted with the parallelism of sidesand is of positive type t -1 1. It is more oonvement, however, toregard this second element on its negative side, when it is aptlycharacterized as 'diversity of directions of sides" 1 . Evidently.

'Soo page

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which the ratio r of the longer side to the shorter is indicated ineach ease.

It can perhaps be justly said that a shape not suggesting asimple numerical ratio like 1 to 1 or 2 to 1 is desirable in manyconnections where the rectangle is used as an element in compo-sition. hi that event the square and the double of a square are

r I r 1.411 r 1.1;1s4

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excluded. .1Ioreover, perhaps the rectangle approved by Platomight. nut serve since it does not differ suffieiently from the doubleof a square. But this still leaves a wide range of choice, embracingthe eases r 1.414 and also r = 1.618. fur instance. Of these twoI slightly prefer the rectangle with ratio 1.414 to the (;olden Rect-angle with ratio 1.618.

NOW itmay he that certain persons, through their acquaintancewith and liking for Greek art, have come to individua.ize andidentify with fair approximation the particular shape embodiedin the Golden Rectangle. For such persons an intuitive associationof purely accidental ellaraeter Nvould he established in favor of the(;olden Rectangle. Only hr assuming that a number of his ex-periinentzd subjects were of this type ran I understand the experi-na.aal results of Fechner in favor of-a particular ratio of nearly8 to 5.

In comparing the square and the rectangle, it should nut heforgotten that the rectangles contain an infinitude of shapes, de-pendent on the ratio of the siffl's, lereaS the :4414141re 111T:,(11tr.1 butat Singh' shape. Hemp the rectangles provide a much more flexibleinstrument in dcsign than the square does. It is. for example,obvious that when a rectangular frame is used for a portrait thesquare shape is in general less suitable than that of a rectanglewith the height greater than the breadth. However. I believe thatthe square is much inure often used than ally other single rev-tanuldar form. such the 11411(14'11 liVrtallide.

rem arks do unit 41111tV do full justice to the special formsof the reetangles when these are not used singly but in combinationwith other polygonal forms. For instance. the arrangement of two

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300 Jatt 0atilm apt iur.110..ta.! 1.1acixa .utu 111.1..mA

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0111m .C110a.lagq)(11 pap.nrzlaa st stxp plpiozpott 0113 lump: S.1330m1Ss 01t3 3.)11J liutpupisttlowx 11:.03.1a. all 3 lda.)xa

3100ERIVAN ILLXIS HILL RLi

POLYGONAL FORMS 179

In this connection we may note the fact that the convex polygonof many sides is more likely to be pleasing than the rei:ntrant one,particularly if the latter contains a diversity of niches.

A second very important new facto(' makes its appearance whenthe polygon is directly related to -3mile uniform network of horizontaland vertical lines, as in the case of the Greek cross [see (al,Figure 8j, or else is closely related to a uniform diamond network

(a) (b)FIGURE 8

see (b) I with its sides equally inclined to the vertical. this dia-mond network in turn sugg,esting a uniform horizontal- verticalnetwork.

L'vidently the aesthetic factor of close relationship to a uniformhorizontal-vertical or diamond network plays a 11111da Went al partand enhances the aesthetic value of many polygons listed, e.g., thesquare (No. I), the rectangle (No, 2) , the diamond (No. 4) , thehexagram (No. 6). the Greek cross (No. 9) . t he swastika emblem(No. 41 , etc. In the case of the square, rectangle, and diamond,

1.!1" validity of this association is perhaps debatable. But, so fre-quently do we see these polyg.ons used in conjunction with a net-work that it seems proper to regard them as suggesting relationshipto a uniform horizontal-vertical or diamond network. On theother hand, it is evidentl not levitini:de to regar.1 the associationwith :t uniform diamond network ns possessing. interest equal totl.at of association with a horizontal-vertical network. It is withtLese facts in mind that the empirical rule dealing with the aestheticfactor of relationship to such a network will be formulated. Theassociational basis of this f:letor in everyday experience is obvious.Systems of lint's placed in the regular array of a network are con-,tantly tort with. and their rehoion,hip to one another is intui-tively apprceiaceti.

Iii the osideration of such more eomplicated pol.ygons itappears a.. .1,at some kltltl of symmetry is always required if the


polygonal form is to be at all attractive. When this requirement isnot met, no degree of relationship to a horizontal-vertical network,for instance, can entirely offset the deficiency. Thus. if symmetryis lacking, the fact appears as a definite negative aesthetic factorwinch must be taken account of.

Evidently a further actual aesthetic factor in many cases isSOME' accidental association, such as is present, for example, inthe case of the cross and the swastika. The mathematical theorytakes no :ireount Of such completely indefinable -elements of order,"although they have a definite aesthetic effect.

9. On the Structure of the Aesthetic Formula. Accordingto the general theory proposed in the lir:4 chapter, we seek anaesthetic formula of the type .11 -_, O r where M is the aestheticmeasure, 0 is the order, and r is the complexity. In the case ofpolygonal form before us, O will he separated into five elements,

0 1- E +I? FThe aesthetic factors encountered above art' correlated in thefollowing way with and these five elements:

C: c,unplcxity.vt-rtivat symmetry (+I.

E: equilibrium (+).R: rotation:11 syto)ertr (-f-).

111 cltla iation In a lioriZoat.il-rilical attwork1': 111)-all4avlory form IllytIlvaa: flino of thr followine famors: too

suLdt (11,1:turf,: front vrtit-r otl,.; sMvs (I, or :Ludestoo no:i 0 or IN0 or ally otlwr :India :ally of form; diversity

eatlio7: ( i; un,u; (ortrii 1:atalit t ) ; diverriity ofthtc!ioas ); lack of smtotry (

It %in be observed that the term F involving the generalattributes of unsatisfactory form is an "(m)Ilium gatherumn for allthe netzative aesthetic factors which have been noted.

The various indifferent fartors of type 101 play 110 part ' Icourse. Some of these art' equality of soles. perpendicularity ofsides, and inclined or horizontal symmetry [without vertical axisof symmetry).

In the course of the teelillical ealoNtion of V, E, Ili-, p,and su of .11. to %%nch we MOW proceed. a simple mathematicaleffileept. IlallIel. that of the group of motions of the given polygon.will be introlluerel. voncept !ous a basic mathematicaladjunct, necessary for the comprehension of the problem before us.

POLYGONAL FORMS

The aesthetic measure M defined by the forniu'a will turn outto depend upon the orientation of the polygon. In case an axisof symmetry exists, the highest aesthetic measure of a particularpolygon will invariably be found when such an axis is taken in thevertical direction. But the same measure will be obtained if thisorientation is reversed. This seems to be in accordance with thefacts which are observed. An apparent exception is furnished by theRoman cross (,No. 49) and certain other polygons, when an inversionappears to diminish notably the aesthetic value. But this will befound to be due to the fact that an important connotative elementof order is thereby removed; for instance, in the case of the crossit is the religious association. Our conclusion, therefore, is thatthese exceptions are apparent rather than real.

10. The Complexity C. The complexity C' of a polygon willhe defined as the least number of indefinitely extended straightlines which contain all the Sider; of the polygon. Thus for anyquadrilateral the complexity is evidently 4; for the Greek cross(, No. 9) the complexity is 8, although the number of sides in theordinary sense is 12; for the pinwheel-shaped figure (.No. 531. thecomplexity is evidently 10; and so on.

The psychological reasonableness of this empirical roe is evi-dent. For COI1VCX polygons, and also for polygons which are notconvex but which dl) not possess any two sides that are situated inthe same straight line, the complexity C is merely the number ofsides. As the eye follows the contour of the polygon in lookingat the various sides in succession, the effort involved would appearto he proportional to the number of sides. On the other hand, ifthere are two or more sides on one and the same straight line.the eve follows these in one motion. For example, in the case ofthe Greek or Roman cross.. the eve might regard it as made up oftwo rectangles. These considerate )ti suggest that the definitionchosen for the complexity is appropriate.

11. The Element V of Vertical Symmetry. The organiza-tion of the entire polygon which results from vertical symmetry isobvious to the eye. By long praetiee we have beeonte accustomedto appreciating symmetry of this sort immediately. On this ac-count the eloment is particularly significant.

Sktil give to the value 1 if the polygon po,st.sses sym-metry about the vertical axis. and the Yalta. 0 in the cant airy case.In other words, the element 1' will be a unit element of order, and,

182 THE SIXTH 'YEARBOOK

since there exist various polygons of pleasing quality, such as theswastika, which do not have vertical symmetry, we shall assignthe value 0 rather than some negative value to V when there is nosuch symmetry.

A large proportion of the polygons listed possess such verticalsymmetry, and it can he verified that the presence of such sym-metry is immediately and favorably recognized.

12. The Element E of quilibrium. We consider next thesecond element E of order concerned with equilibrium. It has beenpreviously observed that when the polygon has vertical symmetryor when it rests upon a suilieiently extended horizontal base, it isfelt to be in equilibrium.

In order to specify completely the requirements for equilibrium,we note first that it is optical equilibrium which is referred to,rather than ordinary mechanical equilibrium. For example, thepinwheel polygon No. 53 would actually be in (unstable) mechaniealequilibrium if turned through an angle of 45°, inasmuch as thecenter of area would lie directly above the lowest point. Never-theless, it does not give the optical impression of equilibrium, sincethe eye is not accustomed to estimating accurately the balance offigures acted upon by gravity.

What is really needed in order that the feeling of completeequilibrium be induced, is either that there be symmetry about thevertical, or that the extreme points of support at the bottom of thepolygon are sufficiently far removed from one another, with thecenter of area ling well between the vertical lines through thesetwo extreme points.

In order to state sufficient requirements for this, we shall agreethat coniplete optical equilibrium will he induced if the center ofarea lies not only between these two lines. but at a distance fromeither of them, at least one-sixth that of the total horizontalbreadth of the polygon. If this somewhat arbitrary condition issatisfied, as well :is in the (`:Ise of vertical symmetry. we shall giveE the value H1. It the polygon does not satisfy these conditionsbut is in equilihritmi in the ordinary mechanical sense, we shalltake E to be 0. Otherwise we shall take E to be 1. inasmuchas the lack of quilibrium is tlu definitely objectionable.

In general the It orientation of a polygon xill be one ofcomplete l'qUiiihrill111. Among, the ninety polygons listed, this isthe ease wit how, exception.

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SiX13 111: 1110(11: 41111 1I0i11:10.1 3011

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0111 jo auuld mil 04 auluolptiadaati Si 110)141:10.1 jo s!xv 011) Salauuti.s pulopt: 30a jo asy.) alp tit 41Iop!sucl 11:p!Ill x41 ()) uo.t.q0(1 0114 salovsal

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`SalauituSs JO SIX!: !Iv 4110(p: cost illino.up 1)a41:40,! jt quip tulaaa Sil 401 tliU spil a)puu (.. a0p.10 UI 403 paziajaa Spuzupt;

A.gatutu.i's pxu alp puu .1!!!.101)1st100 A1011 an: 0.11 liaIliM .C.patutuSs 1uuo13u401 alp ilaimgag luuaultifls Imulaaa Si

4405400 alp 4U S1:A1 4! 0.1011M 04 )1:111(1 SOON lu!oci 5.I3A0 113114 OS 'palaajja uoaq amui ium uopulo.o.! apidtuoa I; lug

ùomsod 11214th! -su 04 alotim 1: 1)atian4aa 0(1 ..quo 400 !!!.m.u0Sind atp iiatull XiaArsr:00ati: papimiaa aq 2u!puo1l5a.t10a alp j't

oogg Jo aos!Atp 401:x0 uv tii tiopim b/c09}; tiopulol JO 11

Spuapt.a Si adaip `Salaiutu.s II:11091;pm sassassod uo!Mio(1 alp ualim asua Dip uj -AalatutuAs reuogeloa ;o zuatuam a4.L ti EST PIVNODAIOd


There is also a second case when the rotational element R isfelt almost equally favorably, despite the lack of true verticalsymmetry; namely, when the minimum convex polygon which in-closes the given polygon does not abut upon any of its niches, and issymmetric about a vertical axis. Polygons Nos. 41, 50, 51, and 69illustrate this case. Here the inclosing convex polygon is sostrongly suggested, with its q axes of symmetry, as to suggestvividly the rotational element.

On the other hand, even though the minimum inclosing polygonis symmetric about a vertical axis, the same effect is not felt if theniches abut on its vertices. This third case is illustrated by poly-gons Nos. 53, 67, 85, 88, 90. In partial explanation of this differ-ence in effect it may be observed that. in the first two eases thecenter of the polygon is clearly defined, either by means of theaxes of symmetry of the given polygon or by means of those of theniiuiniutn enclosing polygon which are strongly suggested. Thiscircumstance operates advantageously.

In both of the first two eases we shall take the element 1? asq/2 so long as q does not exceed 6. For q equal to 6 or greater,we take R = 3, since the effect of the rotational element is limitedand seems to attain its maximum when q is 6. The mathematicalreason for the particular choice of q/2 as the value of 1? whenq does not exceed 6 is alluded to in the next section.

We pass next to the further consideration of those polygonsfalling under the last case. Two possibilities need to be distin-guished according as q is even or odd respectively, thus giving riseto a third and fourth case respectively.

When q is even, there is central symmetry, and such centralsymmetry is appreciated immediately. In particular, it enablesthe observer to fix the center accurately. Here there are no axesof symmetry, actual or suggested. :!Ince it is only these that bringout clearly the rotational symmetry, the rotational symmetry assuch plays only a small role; and so we take I? = 1 in the thirdcase, whatever be the value of q so long as it is even. The justifica-tion from the mathematical point of view of the choice 1? = 1 inthis case will not be attempted here.

Polygons Nos. 39, 45. 48, 53, 65, 67, 74, 77, 84, 85 illustrate thisthird case li = 1.

On the other haml, the four polygons, Nos. 79. 88, 89, and 90,illustrate the convex and rei.'ntrant types in the fourth case. It

POLYGONAL FORMS 185

will be observed that even in the convex type one is scarcely awareof the rotational symmetry.

Thus in this fourth case, as well as in any case where there is norotational symmetry whatever, we are led to take R = 0.

14. The Group of Motions of a Polygon. It has been seenthat either a rotation of a polygon about one of its axes of sym-metry through 180°, or a rotation of the polygon about its centerof area through a multiple of the fundamental angle of rotation3609q, will return the polygon to its original position. There areno other motions which can leave a polygon unaltered. Amongthese motions we will list (by convention) the particular motionwhich moves no point, as well as the other motions A, B, . . . , ifsuch there be.

Definition. of the grout of motions of a polygon. The collec-tion of rotations of a polygon about each of its axes of symmetrythrough 180°, and of the rotations about its center of area throughany angle of rotation, will be termed the "group of motions" ofthe polygon.

The group of motions of a polygon has the fundamental prop-erty th it two such motions A and B performed successively willalso return the polygon to its initial position, and so be equivalentto a single motion C of the group; or symbolically, AB = C.

This leads us to two further related definitions:Definition of conjugate figures. If F be any figure in the plane

of a given polygon, which takes the successive positions F, FA, FR,. . . under the corresponding motions I, A, B, . . . of the group,the figures F, FA, FR, . . . are said to he "conjugate."

Definition. of fundamental region. A region of the plane whichwith its conjugate regions fills the entire plane in which the givenpolygon lies. but in such a way that these regions do not overlap.is said to he a "fundamental region" for the given group of motions.

It is worth while to give an illustration. For the rectangle withvertical axis of symmetry, the motions of the group (besides I)are evidently the rotations S. and .qt/ of 180° about the vertical andhorizontal axes of symmetry respectively, and the rotation I?

through 180° in its plane about the center.It is evident, Om- two opposite sides of the rectangle are con-

jugate under this group of motions. Similarly, the four verticesof the rectangle are conjugate under the group. Finally, it is clear


that any one of the four quadrants into which the plane is dividedby the two axes of e) mmetry Is a fundatnental region for this groupof motions.

It is evident that these concepts are really necessary for theunderstanding of the element It above treated, In particular, thequantity q/2 It 1. 1 for q = 2 when the fundamental region isa half plane, and is inversely proportkml to the angular size ofthe fundamental region. This evaluation of If agrees with the chokeof V as 1, bemuse of the fundamental half plane in the case ofvertical symmetry.

IS. The Element UV of Relation to a Network. In manypolygons of the list, as has ken previously noted, there is obviouslya close relationship of the given polygon to a uniform horizontal-vertical network, and this relation is at once recognized as pleas.ing,

Evidently the corresponding element UV in 0 is connected withcertain motions of the plane in much the same way as the element.V is connected with a motion of rotation about a vertical axis,and the element It with a motion of rotation about a center, Infact, such a uniform network returns to its initial position whenany one of a variety of Inundatory motions of the plane is made,while these same motions will take the polygon to a new fund-tio9 in which it may in large measure have the same boundinglines Its it had in its first position. This happens when most ofthe sides of the polygon coincide with the lines of such a network.In an incomplete way, then, the element //V is connected withny)tions of the plane just as are the elements V and R.

The most fu .orahle case is evidently that in which the polygonhas all its sides upon a uniform network of horizontal and verticallines in such wise that these lines completely fill out a rectangularposition of the network, In this case only do we take //V = 2.polygons Nos, I, 2, 9, 23, 25, 29, 29, etc illustrate this possibility.

The choke of a value //V = 2 seems natural since there areessentially two kinds of independent translatory motions which re-turn the network to its original position; namely, a translation tothe right or left, and a translation up or down, Any other trans-lation may be regarded as derivable by combination from these two.

A similar ease is that in which the sides of the polygon all heupon the lines of a uniform network formed by two sets of parallellines equally inclined to the vatic al, and fill out a diamond-shaped

POLYGONAL FORMS 117

portion of the network. But the effect here is less favorable sothat we take UV = 1,

It becomes necessary at this stage to face a somewhat difficultand vexing question, and assign an index in all cases when thepolygon is agreeably related to a uniform horisontal.vertleal ordiamond network, We shall fix upon the following empiricalrules, We define Ill' to be 1 if the polygon fills out a rectangularportion of a horizontal-vertien1 network in such a way that thefirst set of conditions holds, with the following possible exceptions:one side of the polygon and its conjugates may fall along diagonalsof the rectangular portion or of adjoining rectangles of the net-work; one vertical and one horizontal line, as well as their con-jugates, may nut be occupied by a side of the polygon.

Illustrations of this case //V= I are furnished by polygons Nos,19, 24, 42, 43, 49, ISO, 02, 00 of the list given above.

NV will also be defined to be 1 if the polygon fills out a dia-mond-shaped portion of a diamond network with the followingpossible exceptions: one side of the polygon and its conjugatesmay fall along diagonals of the diamond-shaped portion or of ad-joining cliamonds of the network; one line in the diamond-shapedportion and its conjugates may not be occupied by a side.

Polygons Nos, 5, 0, 24, 53, 05, 08 are illustrations of this ease//V =1,

In every ease when //V is I we shall demand that at least twolines of either set of the network shall be occupied by a side. Inall other eases whatsoever we shall take 11V = O.

It is obvious that the above determination of indices for theelement //V is largely arbitrary. Nevertheless, it seems to agreewith the facts c.hserved,

16. The Element F of General Form. There remains to betreated the factor dealing with general form which we have de-scribed as an "omnium gatherum" of unfavorable elements (Sec-tion 0) ,

The case where is 0 corresponds to satisfactory form. Herethe analysis made in the earlier sections suggests the followingconditions: the minimum distance from any vertex to any othervertex or side must not he too smallfor defit,iteness, we say it isnot to he less than 1/1,) the maximum distance between points ofthe polygon; no ang;e is to he too small or too largefor definite-ness we say not less than 201 nor greater than 1600; the-^ is to be at


most one niche and its conjugates; there is no unsupported refin-trant side; there are at most two directions and their conjugatedirections, provided that horizontal and vertical directions arecounted together as 6, o; there is sufficient symmetry to the extentthat not both V and R are 0,

It is possible to justify this choice of conditions by reference tothe ninety listed polygonal forms, Here we may omit the firsttwo conditions from consideration, since they merely eliminateambiguity of form. The third condition, which admits only onetype of niche, has its origin in the observed fact that only a singleniche may occur, as in the Creek cross (No. 9), without the formbeing thereby im; aired. The fourth condition is justified by thefact already noted that any unsupported retintrant side producesa disagreeable effect. Similarly, if there are three or more dist+ Jttypes of direction, the polygon appears unsatisfactory, unless twoof these are vertical and horizontal, as in No. 43, for instance; thisfact gives rise to the fifth condition. Finally, the sixth and lastof these conditions is obviously to be based upon the fact alreadynoted, that a lack of effective symmetry is not possible with satis-factory form.

By this rule we take account of all the factors of F as thesewere listed in Section 9.

We consider next the case in which one and only one of the sixconditions foils and that to the least possible extent there may beone type of vertex (and conjugates) for which the first conditionfails; or one type of angle for which the second condition fails; ortwo types of niches; or one unsupported rektrant side; or threetypes of direction when vertical and horizontal directions arecounted as the same; or both V and R may be 0. Under these cir-cumstances we take F to he 1.

In every other case we take F to be 2.All the polygons Nos. 1 to 22 inclusive have F= 0, No. 24 is

the first polygon for which lit is 1, because of its having two typesof niches; No. 23 is the first which has one type of unsupportedreentrant sick. The first case for which P is 1 because of diversityof directions is No. 60. The first for which F is 2 is No. 55 whichis also the first for which V and R are 0.

17. Summary. The various elements in the complete formulaIlf=V-1-E+R+HV F

POLYGONAL FORMS 119

have now been defined. The complete definition might be anent-bled in brief form. Let. us, however, merely recapitulate the mainfacts:

V = 1 if there is vertical symmetry (otherwise 0).= 1 if there Is vertical symmetry or the polygon "rests" on

a long enough horizontal side; E = 0 if there is "unstable"equilibrium; l = 1 if k a polygon appears about to fall to oneside or the other,

R ry,.2 if there is rotational symmetry with angle 360°/ryof rotation, and there are axes of symmetry or axes are stronglysuggested (e,g., the swastika, No, 41 ) except. that 11 is taken as 3 forq > 6; in any other case R = 1 If there is central symmetry, andR= 0 if there is not.

//V = 2 if there is complete coincidence of the polygon witha rectangular part of n uniform horisontal-vertical network (e,g.,No, 41) ; 1W = 1 if there is nearly complete relation to such a net-work, or complete or nearly complete relation to a uniform dia-mond network with sides equally inclined to the vertical: 11V is 0otherwise,

= 0 if the form is satisfactory; F = 1 if there is one elementof unsatisfactory form: as one type of unsupported side; or twotypes of niche; or three types of directions when vertical and hori-zontal are counted as only one direction; or when there is no sym-metry t1' and /1' both 0); F:= 2 if there are more than one of theseelements of bad form,

C is the number of straight lines occupied by the sides of thepolygon.

The ninety polygons listed below in order of decreasing Maccording to the empirical formula show the result of a rigid appli-cation of this formula.

It retrains for the reader to determine for himself, by inspectionof the polygons listed or by consideration of still further polygonalforms, whether or not the formula is to be regarded as reasonablysatisfactory.


I 2

,.. .11.16

1.50 1.25

3

1.10

4 5 60 0 *1.00 1.00 1.00

7 43 9

0.90 0.60 0.7a

10

00.71

13

0.62

11

0.03

12

0.6

H 4 ' 15

0 00.08 0.38

hi

0O.3 0.W)

19 20

0

POLYGONAL FORMS 191

17

0.30 0.80

22 23

O 'IfJ-L

0.00

25"

0.30

0.30

26

0.43

28 20

0.40 :Jo

24

0.50

27

0.42

30

0.80

102

31

0.37

34

0.33

37

0.33

0.33

THE SIXTH YEA111100K

32 33

00.37 0.37

35 36

0.33 0.33

438

0.83

39

0.3a

41 42

0.33 0.29

43 43

0.28 0.29 0.29

POLYGONAL FORMS 193

\,746' 47 48

V 0 00.25 0.20 0.25

49 50

Aby9 c?0.25 0.20

52

55

rfl

53

0.50

0.17 0.17 0.17

sic 59 GO

0.17 0.17 0.14

194

61

0.14

THE SIXTH YEARBOOK

62

11.INI11.

0.14

65

zgr0.11 0.11

6s

0.11

67 68 69

0.12 0.11 0.00

70

N0.00

0.00

71

0.00

0.00

aP

0.00

as

0.00

70

IC:Zu.00

79

0.00

82

0.00

POLYGONAL FORMS 195

77

0.00

do

0.00

83

CI70.00

65 66

0.00 0.00

88

-0.10

89

-0.11

81

0.00

8i

0.00

8a

moo

90

-0.17

Documents

11 096 1E17 At rHOR Pec've, W. D., Ed. Mathematics in ... · 1.:1)111()It'4 PREFACE. Tilts Is the sixth or a series of earhook, N10;011111. coloirii of Te:tehers Of Tathematies bevan