Humanistic Mathematics Network Journal

Issue 6 Article 6

5-1-1991

An Historical Approach to Precalculus andCalculusVictor J. KatzUniversity of the District of Columbia

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Recommended CitationKatz, Victor J. (1991) "An Historical Approach to Precalculus and Calculus," Humanistic Mathematics Network Journal: Iss. 6, Article 6.Available at: http://scholarship.claremont.edu/hmnj/vol1/iss6/6

AN HISTORICAL APPROACH TO PRECALCULUS AND CALCULUS

VictorJ. KatzProfessor of Mathematics

Univefs;ry of the District of Columbia

backgrounds in history and literature with their scientificstudies. It also encourages the student at every stage ofhiS/her studies to explore the ramifications of scientificworkas it relates to the world around them. And it seemsto me that the prospective scientists and engineers I amteaching will rrcre than ever need a sense of how theirown highly technical work fits in with the needs of societyas they make decisions which wiD affect the fate of theworld.

A reasonable question at this point is how we can doall this when we can bare ly cover all the material in thesynabus as it is. I do not have a simple answer 10 thatquestion. My experience is that to a large extent, thishistorical approach is more lime-efficient than a morestandard approach and that the historical connectionsdrawn help to motivate and excite the students, enabling

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By an historical approach to malhematics teaching ,I do not mean simply giving the historical background loreaChseparate topic or giving a biographical sketch of thedevelopers of various ideas . I do mean the organizing ofthe desired topics in essentially their historical order ofdevelopment. I do mean discussing the historical moti­vations forthe development of each of these topics, boththose within mathematics and those from other fields . Ialso mean connecting the development of each of themathematical topics with the development of the othersciences and with the other things which were going onin the world . Naturally, I cannol always stick precisely tothe historical record. Many seemingly good ideas led todead ends or to methods which are too difficult for thelevel of these particular courses. And one should makeuse of toclay's technology 01 calculators and computersto perform tedious cofT1)Utations rather than have thestudents repeat their ancestors' hand calculations.Nevertheless, using history as a general guide doesprovide an organizing tool for both precalculus andcalculus which helpsto motivate the students and showsthem that mathematics has atNays been an importantpart of the overall culture, not only in the West but also inother parts of the work:l.

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As a possible answer to this dissatisfaction, and asa new way of organizing these two courses, I haveexperimented overthe past several years with an historicalapproach to both precalculus and caJaJlus-consideredtogether as a four or five term sequence. Not only doesthts approach help integrate the discrete algorithmicmaterial with the continuous analytic mathematics, sincein faCllTlJch of the formerwasdeveloped alongsideot thelatter. but it also helps to introdJce our science andengineering students to the retatcnstrc between math­emat ics and the rest ct our culture. As it stands now,many students are sadly lacking in an awareness of theplace which mathematics occupies in our culture. Theyare interested in the mastery of technique to the exdu­sian olleaming the reasons that the ideas were developedand the use 01 mathematics in the work:l. Andwithout theintellectual content behind the mathematical techniques,it appears that in large measure the students fail to graspeven Ihe techniques. An historical approach to thesecourses helps to provide a solid motivation tor the learningOf mathematics as it ties together much of the students'

As a college teacher of mathematics I receive manynew texts each year in precalculus and calculus, eachone trumpeting its virtues and its new ideas. But a studyof the actual material presented shows that not only arethere few new ideas but that chapter for chapter andalmost section for section each such book is a repeat 01every other one . In fact, I amazed my daughter one dayby telling her the titles of the first ten chapters in thecalculus text she was using. without ever having seen thebOOk itself. Since all such texts are the same. one couldassume that there is a general agreement in the math­ematcarccmrrcnny that there is ere correct way to leachprecalculus and calculus. With all the conferences andposition papers of the past several years, first on thedesirabi lity of discrete mathematics and more recently onthe lean and lively calculus, it appears , however, thatlarge numbers of faculty merrbers are dissatisfied withthe way these courses are presented. As a mattero' fact.the high failure rate in calculus seems to indicate thatstudents too are dissatisfied.

them to dO more wor1( independently. Of course, wecannot neglect the development of techn~1 pl'of~ency

or of problem solv ing skills . But again. both of theseaspects can be set into the histolical context. And if, infact , this approach forces us to drop a few topics in thecurrent syllabus , this will only be in line with currentrecomme ndations in any case .

Let me now describe the mathematics course I havein mind . In the context of the school at which I teach, it isnecessary to beg in with preca lculus material. But itwoukl not be difficu lt to beg in this course at a later po int.As will be clear, vinually all of the standard precalaJlusand calculus topics will be dea~ with, but often in an orderand a conte xt differing trom the standard ones . Thecourse description will erroeasee the nove l aspects ofmy approach and how it better serves the students thanthe usual method .

The course beg ins with a description of the commonmathematical knowledge of various ancient civilizations,namely, basic algebra through quadratic equations, ba­sic geometric formulas, the Pythagorean theorem, andthe calculation of square roots. We discuss the nature ofthese ancient societies, the Babylonian, the Egyptian,the Chinese, and what it means for a certain society to"know~ mathematical ideas. Who in the soc iety knewthese ideas? Why did they know them? What kinds ofproblems dkj the y need to solve? In panicutar, wediscuss the value of pi . Wha t does it mean to approx imatethe ratio of the circumference of a circle to its diameterand how good an approx imat ion is necessary? In thesame mode , we also discuss the square root algorithmand its geometric origins. In this context, an introductorydiscussion of the nature of an algorithm is warrantedalong with a discussion 01 accuracy. A review of thequadratic formu la is also useful along with the Chinesenumerical method of solving quadratic equations.

The major change from these ancient civilizations tothe classical Greek in terms of mathematics was in theintroduction ot deductive proof. Th.Js we deal briel ly withthe nature 01 Greek civiliz ation and its differences fromthose of Egypt and Babylon ia. Then . even though thestudents have generally had some course in geometry, adiscuss ion 01 some salient point s of Euclid'S Elements iswa rranted. Among the topics included are the nature oflogical argument and the idea of an axiomatic system, thebasic triangle and parallelogram theorems of Book I, thecircle theorems of Book Ill , 1he construction of the pen­tagon in Book IV (lor its late r use in trigonometry), thesimilarity results of Book VI, and finally, Euclid'Sresult onthe area 01 a circl e from Book XII.

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CUrious ty, there is no biographical data available onthe autrcr of the wond's most farrous mathematics textexcept a few storie s dating from some 700 years laterthan the time of its writing . Since the text is a compilationof several different strands of Greek mathematics andsince it was written in Alexandria, a city whose inhabit­ants came from many diHerent backgrounds, some his­torians have speeulatedthat itwas written by a committee .Arrong the possibilities lor members 01 this committeewould be native Africans,Jews . and even women.Thoughthis particular idea on the authorship of the Elements ispure speculation , it should be made ci ear to studems thatthough certain groups have traditionally been excludedfrom mathematical knowledge, there were always indioviduals who somehow managed to bud<. the trend andmake their own contributions. Unfortunately, sometimesthe history books ignorethesecontribut ions or possibUities,denying a sense of 'own ership" to large pans of ourpopulation.

Euclid's wor1( is followed in the course by an introduc­tion to conic sectons in terms of sect ions of a cone andin the context of the problem of doubling the cube . Igeneralty stretch the historical record a bit here andinterpret Apollonius' w or1( in tenns of coordinate geom-­etry as Idevebptheequations and some of the elementa rypropernes of these imponant curves. I also introduce thebeg inn ings of mathematical physics in the work ofArchimedes, emphasizing in particular the idea of amathematical mod el.

The idea of a model is further devel oped with thestudy of trigonometry, since that subject originated as amathematical tool for astronomy, which in tum, in theGreek wOnd, was based on the two-sphere model of 1heuniverse. Thus I introduce the students to some ideas inastronomy. us ing the model having the earth fixed in thecenter of the heavens. It is interesting , in fact . to ask thestudents for any evidence that the earth is no t stationary.I introdu ce trigonometry itself in essence as pt olemytreated it, though rathe r than deal w ith his chords I givethe modem ratio definitions of the sine. cosine, andtangent . But the sum, diHerence, and half·angl e Icrmrlasare oene following ptolemy'Sgeometric proofs and theseare used to beg in the actual calcu lation of values 01thetrigonometric funct ions . The values for 30, 45, and 60degrees come from simple right triangle ge ometry , whilethose for 36 and 72 degrees come from Euclid's penta ­gon construction. I think it is importan1for 1he studentsactuall y to pertcrrn some of these calculations them ­selves , so that they learn that the sine function on the irca lculator is not mag ica lly generated nor does it comefrom actually measuring triang les. Arrong the be nefits of,these carcnatere is an appreciat ion of interpolation and

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approximation as well as the realization that the sinefunction is nearty urear for small angles. In fact, it wasthat realization which enabled ptolemy to complete thecalculation of his table. I even recalculate the sine usingradian measure, since, in eHeet, radian measure wasused both in Greeceand in India eartyon . Itwas realizedthat for small angles, using this measure, sine x is verynearly equal to x itself.

The students naturally use their calculators ratherthan the calculated Iab'e in applying trigonometry tosolve problems. These problems irclJde not onty planeproblems but also sphericalones, since it was the latterwhich were most importantto the solving of astronomicalproblems. Wrtha brief disoJssion of some astronomicalconcepts and the introdud ion 01 the if1lX)rtantsphericaltrigonometry fonrulas, the students can easily solveproblems suCh as determining the length 01 daylight on agiven day at a given loCation or the exact direction inwhich the sun rises or sets.

The next major topic, as we move out 01 the Greekperiod. is that of equationsolving. Ibeginthis sectionwitha geometric justification of the quadratic formula takenfrom the work 01 the Arab algebraist al-Khwarizmi. (Thatis, completing the square means exactly what it says.)This justification needs a geometric version of the bino­mial theorem (a+b)2. a2+ 2ab + b2. COntinuing in thisvein, I ask the natural question of how to solve a cubicequation. There are several medieval Arab works whichseek to answer this question, although the answers theygive are probably not what students expect. We there­fore diSQJSS what it means to Msolve- a cubic (or any)equation. The Chinese inthe thirteenth century certainlycould solve such an equation numerically, while theArabs of the same time period knew how to do it geo­metrically. One interesting method to discuss is that otat-Just, who used techniques related to the calculus todecide what types of solutions to a cubic were possible.

To find analgebraicsolution.however.one fT'IJst tumto sixteenth century Italy and the story of Tartaglia andCardano, a story that shouldcertainly be sharedwith thestudents. The studentsshouldalso bemadeawareof theRenaissancebackground here and learn why there wasarenewedurgeinEuropetooo mathematics. Inpartio.Jtar.it was otten the merchants who brought mathematicsbackto Europethroughtheirtravels to Aftica and Asia. Inany case. the algebraic solution of the OJbic begins withthe binomial theorem in degree three and, at least in theSimple cases, is not difficult to understand. I do not givea complete treatment of OJbics. but only enough for the~tudents to get the general idea and see why, in thelfTeducible case,complexnurrtlers arenecessary. Ithen

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can introduce these numbers, as Bontlelli did late in thesixteenthcentury, and use them in solving notonlycubicbut also quadratic equations.

It isnow worthwhile to continuethe studyof solutionsof ecaatcns from other pointsofview.once theCOftlllex.ity of the cubic and. perhaps, the quartic 10fTnJias areunderstood. For exaftllle . I use the workof Descartes10develop the factor and remaindertheorems asweD asthemethods of finding rational solutions to polynomialequations. It no rational solutions exist, the Chinesemethod already disaJsse<t as wenas Newton'S method.itself an adaptation of earlier worK. can be used 10approximate sokJtions. So we return to the notionof analgorithm and by using it can expk)re the idea of con­vergence. Again,eventhoughrrodemcot11Xrtersottwarewill enable any polynomial equationto be solvedI"lJmeri·cally with the touch of a bunon, it is always if1lX)rtant forthe students to understand how the algorithmsbuilt intothe software were developed.

Having already considered the binomial theorem inthe cases n • 2 and n • 3, it is now time to give a moredetailedtreatmentof combinatoricsconcentratingon theworK of Pascal and his medieval Chinese. Arabic, andEuropean predecessors. First, the binomial theorem forany positive integral exponent needs to be developed.Naturally, this is the place to introduce and study thenotion of mathematical induction. A considerationof thebackground and motivation for Pascal's triangle is usefulas an introduction to the ideas of probability and thecalculations of permutations and combinations. A sec­ond major result is the fcrmuta forthe sums of powers ofintegers. This is often associated with Bernoulli,but thebasic ideas date lrom somewhat eartier. A third impcr­tant ideatreated here is the general ideaof arithmeticandgeometric sequences, These latter provide a gentleintroduction to the idea of an infinite series and its sum,an ideaalreadyunderstoodinsomesensebyArchimedes.

Once we have geometric and arithmetic sequences.it is time to investigate logarithms. Idiscuss the scientificneedfor logarithms as anaid in the tediouscomputationsnecessary for the preparation of astronomical tables. SoI also need to diSOJss the need for suCh tables in the ageof European exploration and discovery. In anyevent. therelationshipbetweengeometricandarihmeticseqJencesand the desire to convert fTlJltiplication to addition byusing this relationship leads to the necessity for thechoice of agood base. Adaptirv the ideas of NapierandBriggs. I show the students why -nensar logarithms asweD ascommon k:garithms were both developed. Infact.the question of the -naturalness· of natural Iogarittvnsleads via the worX of Napier himsell to some imponant

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ideas of differential caculus as well as to the idea of theexponential function.

Another idea being developed in the 17th century,through the wol1( of Fermat and Descartes, was that ofanalytic geomet ry. We therefore return to Apollonius'conic sections and show algebraically that any quadraticeqoaton in two variables leads to a conic seocn. Theimportant tangent and focal properties of these curvesare then treated as a 1urther introduct ion to ideas ofcalculus. Keple r's laws are then discussed as a con­tinuation of our eerter not ion of a mathematical rrodel.Of course, whenever one discusses Kepler, one alsomust discuss Ga lilee and the idea that the scientificsuccess of a part icUlar mathematical mdel does notalways mean its acceptance. But we also deal withGalileo as a mathematical physicist as we treat themotion 01 projectiles and the general idea of a 'unction.Though functions have been discussed earner on an adhoc basis, it is here tnat Jmake the first attempt to devek)pthe idea in detail, first in terms of physical phenomenaand then as a pure ly mathematical idea . In particular, weconsider and graph polynomial funct ions and some easyrational funct ions, including one we will use often later,y . 1/(x+1). Andagain the idea of a mathematical modeloccurs, this time in earthly terms rather than in theheavens. Finally,the graphs of the logarithm, exponential,and trigonometric1unctions are briefly discussed, leavingmore details to the development of the calaJlus of thesefunct ions .

Certain calculus ideas having been introduced ear­lier , it isnow time for a detailed discussion, using the work01 Fermat , of the two basic problems of calculus, areasand extrema (or tangents) . I first solve these two prob­lems for the curves y • x2 and y • x3 and then proceed togeneralize to v>xn. For derivatives, I use the binomialtheorem, and lor integrals, I use either the material onBernoulli numbers or the work on geometric sequences.In both cases, we need Ihe basic idea of a limit, so anintuiHve discussion of limits is warranted here. Theelementary resuns on power funct ions can then be ex­tended to a procedure for find ing der ivatives and integralsof polynomials. Now anhough Fermat essentia lly hadthese results, he never not iced what Newton did, theFundamental Theorem. In any case, we have nowreached the central figure in the scientif ic revolut ion . Astatement and at least a sketch of a proof of the Fun­damental Theorem can now be given, prObably in terms01 velocity and distance . In fact , with the right prepara­tion, the students can "discove r" this theorem lor them-­selves. But I also discuss to some extent the scientificrevolution itself and Newton's part in it. In particular,throughOu1the remainder of the course I deal with some

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Of the problems the calcu lus enabled Newton and othersto solve.

It is at this point in the syllabuS that I believe thehistorical aporoacn makes its most important contribu ­tion10the study of calculus. namely the earty introcluctionof the notion of a power series . Power series were oneof Newton's ear1iest discoveries in calculus and onewhich he used constantly. In fact . I also note that for thesine and cosine, power series had been deve loped evenear1ier in India. It is qu ite effective pedagogica lly tointroduce series atthe ear1iest possible moment,name ly,as soon as the basic derivative and integral algorithmsare know n and the fundamental theorem is proved.Power series can then be used as a theme throughout therest of the course. They provide examples of algorithms.expl icit calculations of certain integrals, ideas on therelation ships among various functions. a further intro­duct ion 10 the fundamental idea of convergence, and aprime exarrple of the melhod of discovery throug hanalogy.

As a beginning 10 the study of power series, wesimply deal with them as generalized polynomialSwhichwe can add , subtract , rrultiply, and divide , We thendiscuss convergence by trying to use power series torepresent funct ions . In particular, we can think 01 powerseries as generalizing infinite decimal expansions 01numbers. As a first nice exarrple of a power series. Ishow the generalization of the binomial theorem tonegative and fractional exponents. I then use this tocalculate square and cube roots. for example. I also notethat , since the power series can be considered as gen ­eralized polynomial functions ,we can take the derivativeand anti-derivative term by term .

We do need other techniques in caicnus . So weproceed tctne bas icapproximalion theorem ,f(x+h). f{xl+ 1'(x)h and then the ideas of leibniz and his followers, inparticular the idea of a differential. The approximationtheorem gives us straightforward proofs of the productrule, quotient rule, and cha in rule. We can then dea leasily with integrat ion by parts and by substitut ion. On

the other hand , we can integrate Y1 - )(2 and~ byusing power series since the other methods do not apply.Ofcourse ,physicatproblems, partcuianytbcse centeredaround Newton's laws ot motion, must be dealt with herenowthat thebasic tools are available. Maximum-minirrumproblems are among the more important of these typesof problems.

As in any ca lculus course, it is now necessary to dealwith the transcendental functions. The natural logarithm

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can be developed as is standard, but is also historicallyjustified, via the integral of 1/(1+x). But by consideringthe power series for that fundion, we can also get theseries for the logarithm. And this, 01 course, enables usto calculate values. The series for the exponentialfunct ion can then be developed and disaJssedby invertingthe series for the logarithm, by Euler's method of usingthe binomial theorem, or, pertlaps, by solving the differ·ential equa tion v> kyoessentia lly introduced in the earliertreatment ot these functions. There are many interestingproblems involving the exponential and logarithm func ­lions in earty calculus textbooks. It is interesting fortoday's students to see what kinds of problems eartierstudents solved, so I show them some from the calculustext of Maria Agnesi. probably the best of those before theworks of Euler and evidence that women could and didstudy mathematics.

The calculus of the trigonometric functions isdevel-­oped via Newton's idea of relating them 10arcs of circles .This leads again to the "naturalness· of radian measure.Thus I begin with arc length and the definition ot theinverse functions. Again, these are done in terms ofpower series. The series for sine, cosine, and tangentcan easily be developed by various methods as can thebasic rules for derivatives. In fact. the latter is best doneby a geometric argument using differentials rather thanvia the limit argument commonly used. The powerseriesrepresentations of eX. sin x, and cos x then provide anatural question of how these functions are related. Theanswer leads into a renewed discussion of corrorexoombers and also into a treatment of the hyperbolicfunctions. Finally, the notion of simple harmonic m:>tionand its associated differential equation y" • -ky is dealtwith and shown to result in the trigonometric functions.Thus the basic periodicity of these functions can beunderstood in terms of an important physical idea.

Other types of physical problems are treated in theprocess of dea ling with various techn~esofintegration.Not too much time is spent on the techniques them-

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selves , however, since power series methods are avail­able to give results and since new computer algebrasystems will generate these results in any case. But wedo need to deal with such ideas as arc length, volume.anc:l center of gravity and see what integrals are neces­sary to solve these problems. At the same time, some ofthe elementary ideas of differential equations. includingthe separation of variables and the integrahon of exactequations, are covered in the context of the physieal andmathematical problems which led to their study in the firstplace.

It is at the end of the one-variable section of thecalculus course that I give a detailed treatment of thenotions of limit and convergence. tt is only after theexperience of dealing with these ideas on an intuitivebasis for many months that J can expect the students tounderstand their theoretical underpinnings and the useof epsilons and dertas. Atter aD. the work of Cauchy andBolzanoocO,medful1y 150 years after that of Newton andLeibna.

An historical approach to the study of precalculUSand calculuscan providevaluable insights to the students.With appropriate examples, it can also serve to show thaIwomen and minorities have been involved in mathematicsin the past and can certainly expect to make furthercontributions in the future . I am convinced that tNsapproach to this ilJllOrtant segment of mathematics isbetterjhan the OJrrentmethod which, in ca tculus at least.begins ~ite unhistoncally, and quite unsuccessfully,with the abstract notion of a limit. Unfortunately, thepublishers of mathematics texts hesitate to publish a textwhich deviates in any major respect from the currentmodel. It istherefore not very easy to get new curriculumideasoUl into the mathematicscommunity, lithe readersof this essay try out some of the ideas expressed in it,however, and exchange their findings, the mathematicalcomrrunity as a whokt may eventuaUysee the benefits 01this approach to the leaching of the most poputarcoursesin the mathematics curriculum

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