THE AMERICAN MATHEMATICALMONTHLYVOLUME 118, NO. 3 MARCH 2011195 Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2011 toJoseph A. Gallian for Distinguished Service to MathematicsBarbara Faires198 Was Cantor Surprised?Fernando Q. Gouv ea210 On Legendres Work on the Law of Quadratic ReciprocitySteven H. Weintraub217 Equimodular Polynomials and the Tritangency Theoremsof Euler, Feuerbach, and GuinandAlexander Ryba and Joseph Stern229 The Sharkovsky Theorem: A Natural Direct ProofKeith Burns and Boris Hasselblatt245 A First Look at Differential AlgebraJohn H. Hubbard and Benjamin E. LundellNOTES262 Lines of Best Fit for the Zeros and for the Critical Pointsof a PolynomialGrant Keady264 Regular Matchstick GraphsSascha Kurz and Rom Pinchasi268 A Recursive Scheme for Improving the Original Rate ofConvergence to the EulerMascheroni ConstantEdward Chlebus275 PROBLEMS AND SOLUTIONSREVIEWS283 Logical Labyrinths. By Raymond M. SmullyanChristopher C. LearyAn Ofcial Publication of the Mathematical Association of AmericaRalph P. BoasSecond Edition revised by Harold P. Boas An ideal choice for a frst course in complex analysis, this book can be used either as a classroom text or for independent study. Written in an informal style by a master expositor, the book distills more than half a century of experience with the subject into a lucid, engaging, yet rigorous account. The book reveals both the power of complex analysis as a tool for applications and the intrinsic beauty of the subject as a fundamental part of pure mathematics. Written at the level of courses commonly taught in American universities to seniors and beginning graduate students, the book is suitable for readers acquainted with advanced calculus or introductory real analysis. The treatment goes beyond the standard material of power series, Cauchys theorem, residues, conformal mapping, and harmonic functions by including accessible discussions of many intriguing topics that are uncommon in a book at this level. Readers will encounter notions ranging from Landaus notation to overconvergent series to the Phragmn-Lindelf theorem. The fexibility aforded by the supplementary topics and applications makes the book adaptable either to a short, one-term course or to a comprehensive, full-year course.Each topic is discussed in a typical, commonly encountered situation rather than in the most general, abstract setting. There are no numbered equations. Numerous exercises interspersed in the text encourage readers to test their understanding of new concepts and techniques as they are presented. Detailed solutions of the exercises, included at the back of the book, both serve as models for students and facilitate independent study. Supplementary exercises at the ends of sections, not solved in the book, provide an additional teaching tool.This second edition of Invitation to Complex Analysis has been painstakingly revised by the authors son, himself an award-winning mathematical expositor. Catalog Code: ICA Series: Textbooks ISBN: 978-0-88385-764-9Hardbound, 2010 List: $63.95 Member Price: $50.95Invitation to Complex AnalysisTHE AMERICAN MATHEMATICALMONTHLYVOLUME 118, NO. 3 MARCH 2011EDITORDaniel J. VellemanAmherst CollegeASSOCIATE EDITORSWilliam AdkinsLouisiana State UniversityDavid AldousUniversity of California, BerkeleyRoger AlperinSan Jose State UniversityAnne BrownIndiana University South BendEdward B. BurgerWilliams CollegeScott ChapmanSam Houston State UniversityRicardo CortezTulane UniversityJoseph W. DaubenCity University of New YorkBeverly DiamondCollege of CharlestonGerald A. EdgarThe Ohio State UniversityGerald B. FollandUniversity of Washington, SeattleSidney GrahamCentral Michigan UniversityDoug HensleyTexas A&M UniversityRoger A. HornUniversity of UtahSteven KrantzWashington University, St. LouisC. Dwight LahrDartmouth CollegeBo LiPurdue UniversityJeffrey NunemacherOhio Wesleyan UniversityBruce P. PalkaNational Science FoundationJoel W. RobbinUniversity of Wisconsin, MadisonRachel RobertsWashington University, St. LouisJudith RoitmanUniversity of Kansas, LawrenceEdward ScheinermanJohns Hopkins UniversityAbe ShenitzerYork UniversityKaren E. SmithUniversity of Michigan, Ann ArborSusan G. StaplesTexas Christian UniversityJohn StillwellUniversity of San FranciscoDennis StoweIdaho State University, PocatelloFrancis Edward SuHarvey Mudd CollegeSerge TabachnikovPennsylvania State UniversityDaniel UllmanGeorge Washington UniversityGerard VenemaCalvin CollegeDouglas B. WestUniversity of Illinois, Urbana-ChampaignEDITORIAL ASSISTANTNancy R. 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Permission to make copies of individual arti-cles, in paper or electronic form, including postingon personal and class web pages, for educationaland scientic use is granted without fee providedthat copies are not made or distributed for protor commercial advantage and that copies bear thefollowing copyright notice: [Copyright the Mathe-matical Association of America 2011. All rights re-served.] Abstracting, with credit, is permitted. Tocopy otherwise, or to republish, requires specicpermission of the MAAs Director of Publications andpossibly a fee. Periodicals postage paid at Washing-ton, DC, and additional mailing ofces. Postmaster:Send address changes to the American Mathemati-cal Monthly, Membership/Subscription Department,MAA, 1529 Eighteenth Street, N.W., Washington, DC,20036-1385.Yueh-Gin Gung and Dr. Charles Y. HuAward for 2011 to Joseph A. Gallian forDistinguished Service to MathematicsBarbara FairesJoe Gallian.The two themes that run through Joseph A. Gallians service to mathematics are (a) en-couraging young mathematicians and helping them to develop successful careers and(b) communicating mathematics to the widest possible audience. He was one of theearly proponents of undergraduates conducting mathematical research, and his REUat Duluth, which began in 1977, is widely regarded as the premier REU. The qualityof the work at Joes REU is evidenced by the 160 papers by participants that grew outof their REU work. These papers have appeared in such journals as Crelles Journal,Journal of Algebra, Journal of Combinatorial Theory, Discrete Mathematics, AppliedDiscrete Mathematics, Annals of Discrete Mathematics, and Journal of Graph Theory.The REU, along with Joes continuing contact with its participants, makes an importantcontribution to developing the next generations of mathematicians. Participants haveincluded some prominent mathematicians, whose careers this REU helped to shape.Joe is also an inspiration to a generation of mathematicians who involve students inhigh-quality undergraduate research in mathematics. Not only is Joe successful withhis own REU, but he is generous with his time and advice to help others to set upREUs. In 2002, Joe was recognized by the Council on Undergraduate Research withtheir Fellow Award, given to members who have demonstrated sustained excellence inresearch with undergraduates.doi:10.4169/amer.math.monthly.118.03.195March 2011] YUEH-GIN GUNG AND DR. CHARLES Y. HU AWARD FOR 2011 195Project NExT is the MAAs widely acclaimed professional development opportu-nity for new or recent Ph.D.s. Joe has been involved with Project NExT since its rstsummer in 1994 when he gave the closing address. The address was so extraordi-narily successful that he has given each subsequent closing address. Later, when Joebecame co-director of Project NExT in 1998, he assumed primary responsibility formany parts of the program, participated in developing the workshop program, and of-ten drafted articles for Focus and reports to the Board of Governors. His boundlessenergy, enthusiasm, mathematical sophistication, and academic savvy have made himthe perfect person to work with the hundreds of new mathematics faculty who havebecome Project NExT Fellows.Joes Project NExT work illustrates that his service to mathematics ranges acrossall the levels of work that needs to be done. Not only does Joe participate in long-range planning and vision discussions, but he also does the small tasks that keep aprogram functioning successfully. As with his REU, Joe does not treat Project NExTas a job for which he has specied, narrowly dened duties, but as a program to whichhe generously gives his time and to whose success he is committed.In his talks, Joe combines thorough preparation, imaginative presentation, and ashowmans air with solid mathematical content. An indication of Joes success atcommunicating mathematics was a standing ovation from the audience at his Pi MuEpsilon Frame Lecture. This audience included high school students as well as profes-sors, and all understood and were excited with Joes talk. Joe has given 24 addressesat national meetings, 65 at MAA Section meetings, and over 200 at colleges and uni-versities. Joe also communicates mathematics beyond the mathematical community.Articles about his work have appeared in twenty-ve news outlets in the United Statesas well as in Europe and India. Four of these were in Science News and one in the NewYork Times. In addition to this he has more than a 100 articles in mathematical jour-nals and other publications including Math Horizons, the Macmillan Encyclopedia ofChemistry, and the Mathematical Intelligencer. Joe Gallian was named by a Duluthnewspaper as one of the 100 Great Duluthians of the 20th Century.Joe has served professional organizations and the mathematical community atlarge. Joe has been national coordinator for Mathematics Awareness Month (2003 and2010); he has served on more than 50 national committees, chairing at least 10 ofthem; he was a Council on Undergraduate Research Councilor for 11 years, servingas chair of the mathematics and computer science division for part of that time; he hasserved as associate editor for Mathematics Magazine and the American MathematicalMonthly; and he has been director or co-director of ve conferences. Joe has refereedfor 40 journals and is a reviewer for NSF, the Research Council of Canada, and theAustralian Research Council. Those who work with Joe know that he is always anactive contributor to a project in which he is involvedhe is efcient and he movesthe project along.Joe Gallians many awards and honors attest to his passion to serve undergraduates,professional organizations, and the mathematical community. He has been honoredwith teaching awards from the University of Minnesota Duluth, the Carnegie Founda-tion for the Advancement of Teaching, and the Mathematical Association of America(Haimo Award). Joe has received the MAAs Trevor Evans and Carl B. AllendoerferAwards and has been an MAA Polya Lecturer. Joe served as second vice president andthen president of the MAA.Joe completed his undergraduate degree at Slippery Rock University, M.A. at theUniversity of Kansas, and Ph.D. at Notre Dame. He is a professor of mathematics andstatistics at the University of Minnesota Duluth, where he was recognized in December2009 with the Chancellors Award for Distinguished Research.196 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118About the Gung-Hu Award. The Yueh-Gin Gung and Dr. Charles Y. Hu Award forDistinguished Service to Mathematics is the successor to the Associations Award forDistinguished Service to Mathematics, rst presented in 1962. It is intended to be themost prestigious award for service offered by the Association. The initial endowmentwas contributed by husband and wife Dr. Charles Y. Hu and Yueh-Gin Gung. Dr. Huand Yueh-Gin Gung were not mathematicians, but rather a professor of geography atthe University of Maryland and a librarian at the University of Chicago, respectively.They contributed generously to our discipline because, as they wrote, We alwayshave high regard and great respect for the intellectual agility and high quality of mindof mathematicians and consider mathematics as the most vital eld of study in thetechnological age we are living in.The Essence of Mathematics . . .The essence of mathematics is accuracy.Walter Bagehot, The Works of Walter Bagehot, Vol. V,The Travelers Insurance Company, Hartford, CT, 1891, p. 457.The essence of mathematics is exact truthfulness.Ellery W. Davis, The Condition of Secondary Mathematical InstructionWith Some Hints to Remedies, Mathematical Supplementof School Science, Vol. 1, No. 1, 1903, p. 8.The essence of mathematics lies precisely in its freedom.Georg Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten,Mathematische Annalen 21 (1883) 564.March 2011] YUEH-GIN GUNG AND DR. CHARLES Y. HU AWARD FOR 2011 197Was Cantor Surprised?Fernando Q. Gouv eaAbstract. We look at the circumstances and context of Cantors famous remark, I see it, butI dont believe it. We argue that, rather than denoting astonishment at his result, the remarkpointed to Cantors worry about the correctness of his proof.Mathematicians love to tell each other stories. We tell them to our students too, andthey eventually pass them on. One of our favorites, and one that I heard as an under-graduate, is the story that Cantor was so surprised when he discovered one of his the-orems that he said I see it, but I dont believe it! The suggestion was that sometimeswe might have a proof, and therefore know that something is true, but nevertheless stillnd it hard to believe.That sentence can be found in Cantors extended correspondence with Dedekindabout ideas that he was just beginning to explore. This article argues that what Can-tor meant to convey was not really surprise, or at least not the kind of surprise thatis usually suggested. Rather, he was expressing a very different, if equally familiar,emotion. In order to make this clear, we will look at Cantors sentence in the contextof the correspondence as a whole.Exercises in myth-busting are often unsuccessful. As Joel Brouwer says in his poemA Library in Alexandria,. . . And so history gets writtento prove the legend is ridiculous. But soon the legendreplaces the history because the legend is more interesting.Our only hope, then, lies in arguing not only that the standard story is false, but alsothat the real story is more interesting.1. THE SURPRISE. The result that supposedly surprised Cantor was the fact thatsets of different dimension could have the same cardinality. Specically, Cantorshowed (of course, not yet using this language) that there was a bijection between theinterval I = [0, 1] and the n-fold product In= I I I .There is no doubt, of course, that this result is surprising, i.e., that it is counter-intuitive. In fact Cantor said so explicitly, pointing out that he had expected somethingdifferent. But the story has grown in the telling, and in particular Cantors phrase aboutseeing but not believing has been read as expressing what we usually mean when wesee something happen and exclaim Unbelievable! What we mean is not that weactually do not believe, but that we nd what we know has happened to be hard tobelieve because it is so unusual, unexpected, surprising. In other words, the idea is thatCantor felt that the result was hard to believe even though he had a proof. His phrasehas been read as suggesting that mathematical proof may engender rational certaintywhile still not creating intuitive certainty.The story was then co-opted to demonstrate that mathematicians often discoverthings that they did not expect or prove things that they did not actually want to prove.For example, here is William Byers in How Mathematicians Think:doi:10.4169/amer.math.monthly.118.03.198198 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Cantor himself initially believed that a higher-dimensional gure would have alarger cardinality than a lower-dimensional one. Even after he had found the ar-gument that demonstrated that cardinality did not respect dimensions: that one-,two-, three-, even n-dimensional sets all had the same cardinality, he said, I seeit, but I dont believe it. [2, p. 179]Did Cantors comment suggest that he found it hard to believe his own theoremeven after he had proved it? Byers was by no means the rst to say so.Many mathematicians thinking about the experience of doing mathematics havefound Cantors phrase useful. In his preface to the original (1937) publication of theCantor-Dedekind correspondence, J. Cavaill` es already called attention to the phrase:. . . these astonishing discoveriesastonishing rst of all to the author himself:I see it but I dont believe it at all,1he writes in 1877 to Dedekind about one ofthem, these radically new notions . . . [14, p. 3, my translation]Notice, however, that Cavaill` es is still focused on the description of the result assurprising rather than on the issue of Cantors psychology. It was probably JacquesHadamard who rst connected the phrase to the question of how mathematicians think,and so in particular to what Cantor was thinking. In his famous Essay on the Psychol-ogy of Invention in the Mathematical Field, rst published in 1945 (only eight yearsafter [14]), Hadamard is arguing about Newtons ideas:. . . if, strictly speaking, there could remain a doubt as to Newtons example,others are completely beyond doubt. For instance, it is certain that Georg Cantorcould not have foreseen a result of which he himself says I see it, but I do notbelieve it. [10, pp. 6162].Alas, when it comes to history, few things are certain.2. THE MAIN CHARACTERS. Our story plays out in the correspondence betweenRichard Dedekind and Georg Cantor during the 1870s. It will be important to knowsomething about each of them.Richard Dedekind was born in Brunswick on October 6, 1831, and died in the sametown, now part of Germany, on February 12, 1916. He studied at the University ofG ottingen, where he was a contemporary and friend of Bernhard Riemann and wherehe heard Gauss lecture shortly before the old mans death. After Gauss died, LejeuneDirichlet came to G ottingen and became Dedekinds friend and mentor.Dedekind was a very creative mathematician, but he was not particularly ambitious.He taught in G ottingen and in Zurich for a while, but in 1862 he returned to his hometown. There he taught at the local Polytechnikum, a provincial technical university. Helived with his brother and sister and seemed uninterested in offers to move to moreprestigious institutions. See [1] for more on Dedekinds life and work.Our story will begin in 1872. The rst version of Dedekinds ideal theory had ap-peared as Supplement X to Dirichlets Lectures in Number Theory (based on actuallectures by Dirichlet but entirely written by Dedekind). Also just published was oneof his best known works, Stetigkeit und Irrationalzahlen (Continuity and IrrationalNumbers; see [7]; an English translation is included in [5]). This was his account ofhow to construct the real numbers as cuts. He had worked out the idea in 1858, butpublished it only 14 years later.1Cavaill` es misquotes Cantors phrase as je le vois mais je ne le crois point.March 2011] WAS CANTOR SURPRISED? 199Georg Cantor was born in St. Petersburg, Russia, on March 3, 1845. He died inHalle, Germany, on January 6, 1918. He studied at the University of Berlin, where themathematics department, led by Karl Weierstrass, Ernst Eduard Kummer, and LeopoldKronecker, might well have been the best in the world. His doctoral thesis was on thenumber theory of quadratic forms.In 1869, Cantor moved to the University of Halle and shifted his interests to thestudy of the convergence of trigonometric series. Very much under Weierstrasss in-uence, he too introduced a way to construct the real numbers, using what he calledfundamental series. (We call them Cauchy sequences.) His paper on this construc-tion also appeared in 1872.Cantors lifelong dream seems to have been to return to Berlin as a professor, butit never happened. He rose through the ranks in Halle, becoming a full professor in1879 and staying there until his death. See [13] for a short account of Cantors life.The standard account of Cantors mathematical work is [4].Cantor is best known, of course, for the creation of set theory, and in particular forhis theory of transnite cardinals and ordinals. When our story begins, this was mostlystill in the future. In fact, the birth of several of these ideas can be observed in thecorrespondence with Dedekind. This correspondence was rst published in [14]; wequote it from the English translation by William Ewald in [8, pp. 843878].3. ALLOW ME TO PUT A QUESTION TO YOU. Dedekind and Cantor met inSwitzerland when they were both on vacation there. Cantor had sent Dedekind a copyof the paper containing his construction of the real numbers. Dedekind responded, ofcourse, by sending Cantor a copy of his booklet. And so begins the story.Cantor was 27 years old and very much a beginner, while Dedekind was 41 and atthe height of his powers; this accounts for the tone of deference in Cantors side ofthe correspondence. Cantors rst letter acknowledged receipt of [7] and says that myconception [of the real numbers] agrees entirely with yours, the only difference beingin the actual construction. But on November 29, 1873, Cantor moves on to new ideas:Allow me to put a question to you. It has a certain theoretical interest for me, butI cannot answer it myself; perhaps you can, and would be so good as to write meabout it. It is as follows.Take the totality of all positive whole-numbered individuals n and denote itby (n). And imagine, say, the totality of all positive real numerical quantities xand designate it by (x). The question is simply, Can (n) be correlated to (x) insuch a way that to each individual of the one totality there corresponds one andonly one of the other? At rst glance one says to oneself no, it is not possible, for(n) consists of discrete parts while (x) forms a continuum. But nothing is gainedby this objection, and although I incline to the view that (n) and (x) permit noone-to-one correlation, I cannot nd the explanation which I seek; perhaps it isvery easy.In the next few lines, Cantor points out that the question is not as dumb as it looks,since the totality pq

of all positive rational numbers can be put in one-to-one cor-respondence with the integers.We do not have Dedekinds side of the correspondence, but his notes indicate thathe responded indicating that (1) he could not answer the question either, (2) he couldshow that the set of all algebraic numbers is countable, and (3) that he didnt think thequestion was all that interesting. Cantor responded on December 2:200 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118I was exceptionally pleased to receive your answer to my last letter. I put myquestion to you because I had wondered about it already several years ago, andwas never certain whether the difculty I found was subjective or whether it wasinherent in the subject. Since you write that you too are unable to answer it, I mayassume the latter.In addition, I should like to add that I have never seriouslyoccupied myself with it, because it has no special practical interest for me. AndI entirely agree with you when you say that for this reason it does not deservemuch effort. But it would be good if it could be answered; e.g., if it could beanswered with no, then one would have a new proof of Liouvilles theorem thatthere are transcendental numbers.Cantor rst concedes that perhaps it is not that interesting, then immediately pointsout an application that was sure to interest Dedekind! In fact, Dedekinds notes indi-cate that it worked: But the opinion I expressed that the rst question did not deservetoo much effort was conclusively refuted by Cantors proof of the existence of tran-scendental numbers. [8, p. 848]These two letters are fairly typical of the epistolary relationship between the twomen: Cantor is deferential but is continually coming up with new ideas, new questions,new proofs; Dedekinds role is to judge the value of the ideas and the correctness ofthe proofs. The very next letter, from December 7, 1873, contains Cantors rst proofof the uncountability of the real numbers. (It was not the diagonal argument; see [4]or [9] for the details.)4. THE SAME TRAIN OF THOUGHT. . . Cantor seemed to have a good sensefor what question should come next. On January 5, 1874, he posed the problem ofhigher-dimensional sets:As for the question with which I have recently occupied myself, it occurs to methat the same train of thought also leads to the following question:Can a surface (say a square including its boundary) be one-to-one correlatedto a line (say a straight line including its endpoints) so that to every point of thesurface there corresponds a point of the line, and conversely to every point of theline there corresponds a point of the surface?It still seems to me at the moment that the answer to this question is verydifcultalthough here too one is so impelled to say no that one would like tohold the proof to be almost superuous.Cantors letters indicate that he had been asking others about this as well, and thatmost considered the question just plain weird, because it was obvious that sets ofdifferent dimensions could not be correlated in this way. Dedekind, however, seems tohave ignored this question, and the correspondence went on to other issues. On May18, 1874, Cantor reminded Dedekind of the question, and seems to have received noanswer.The next letter in the correspondence is from May, 1877. The correspondence seemsto have been reignited by a misunderstanding of what Dedekind meant by the essenceof continuity in [7]. On June 20, 1877, however, Cantor returns to the question ofbijections between sets of different dimensions, and now proposes an answer:. . . I should like to know whether you consider an inference-procedure that I useto be arithmetically rigorous.The problem is to show that surfaces, bodies, indeed even continuous struc-tures of dimensions can be correlated one-to-one with continuous lines, i.e.,March 2011] WAS CANTOR SURPRISED? 201with structures of only one dimensionso that surfaces, bodies, indeed evencontinuous structures of dimension have the same power as curves. This ideaseems to conict with the one that is especially prevalent among the represen-tatives of modern geometry, who speak of simply innite, doubly, triply, . . . ,-fold innite structures. (Sometimes you even nd the idea that the innity ofpoints of a surface or a body is obtained as it were by squaring or cubing theinnity of points of a line.)Signicantly, Cantors formulation of the question had changed. Rather than askingwhether there is a bijection, he posed the question of nding a bijection. This is, ofcourse, because he believed he had found one. By this point, then, Cantor knows theright answer. It remains to give a proof that will convince others. He goes on to explainhis idea for that proof, working with the -fold product of the unit interval with itself,but for our purposes we can consider only the case = 2.The proof Cantor proposed is essentially this: take a point (x, y) in [0, 1] [0, 1],and write out the decimal expansions of x and y:(x, y) = (0.abcde . . . , 0. . . . ).Some real numbers have more than one decimal expansion. In that case, we alwayschoose the expansion that ends in an innite string of 9s. Cantors idea is to map (x, y)to the point z [0, 1] given byz = 0.abc de . . .Since we can clearly recover x and y from the decimal expansion of z, this gives thedesired correspondence.Dedekind immediately noticed that there was a problem. On June 22, 1877 (onecannot fail to be impressed with the speed of the German postal service!), he wroteback pointing out a slight problem which you will perhaps solve without difculty.He had noticed that the function Cantor had dened, while clearly one-to-one, was notonto. (Of course, he did not use those words.) Specically, he pointed out that suchnumbers asz = 0.120101010101 . . .did not correspond to any pair (x, y), because the only possible value for x is0.100000 . . . , which is disallowed by Cantors choice of decimal expansion. Hewas not sure if this was a big problem, adding I do not know if my objection goes tothe essence of your idea, but I did not want to hold it back.Of course, the problem Dedekind noticed is real. In fact, there are a great many realnumbers not in the image, since we can replace the ones that separate the zeros withany sequence of digits. The image of Cantors map is considerably smaller than thewhole interval.Cantors rst response was a postcard sent the following day. (Can one envision himreading the letter at the post ofce and immediately dispatching a postcard back?) Heacknowledged the error and suggested a solution:Alas, you are entirely correct in your objection; but happily it concerns only theproof, not the content. For I proved somewhat more than I had realized, in that Ibring a system x1, x2, . . . , x of unrestricted real variables (that are 0 and 1)into one-to-one relationship with a variable y that does not assume all values of202 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118that interval, but rather all with the exception of certain y

. However, it assumeseach of the corresponding values y

only once, and that seems to me to be theessential point. For now I can bring y

into a one-to-one relation with anotherquantity t that assumes all the values 0 and 1.I am delighted that you have found no other objections. I shall shortly writeto you at greater length about this matter.This is a remarkable response. It suggests that Cantor was very condent that hisresult was true. This condence was due to the fact that Cantor was already thinkingin terms of what later became known as cardinality. Specically, he expects that theexistence of a one-to-one mapping from one set A to another set B implies that thesize of A is in some sense less than or equal to that of B.Cantors proof shows that the points of the square can be put into bijection with asubset of the interval. Since the interval can clearly be put into bijection with a subsetof the square, this strongly suggests that both sets of points are the same size, or,as Cantor would have said it, have the same power. All we need is a proof that thepowers are linearly ordered in a way that is compatible with inclusions.That the cardinals are indeed ordered in this way is known today as the Schroeder-Bernstein theorem. The postcard shows that Cantor already knew that the Schroeder-Bernstein theorem should be true. In fact, he seems to implicitly promise a proof ofthat very theorem. He was not able to nd such a proof, however, then or (as far as Iknow) ever.His fuller response, sent two days later on June 25, contained instead a completelydifferent, and much more complicated, proof of the original theorem.I sent you a postcard the day before yesterday, in which I acknowledged thegap you discovered in my proof, and at the same time remarked that I am ableto ll it. But I cannot repress a certain regret that the subject demands morecomplicated treatment. However, this probably lies in the nature of the subject,and I must console myself; perhaps it will later turn out that the missing portionof that proof can be settled more simply than is at present in my power. But sinceI am at the moment concerned above all to persuade you of the correctness ofmy theorem. . . I allow myself to present another proof of it, which I found evenearlier than the other.Notice that what Cantor is trying to do here is to convince Dedekind that his theoremis true by presenting him a correct proof.2There is no indication that Cantor had anydoubts about the correctness of the result itself. In fact, as we will see, he says sohimself.Lets give a brief account of Cantors proof; to avoid circumlocutions, we will ex-press most of it in modern terms. Cantor began by noting that every real number xbetween 0 and 1 can be expressed as a continued fractionx = 1a + 1b + 1c + 2Cantor claimed he had found this proof before the other. I nd this hard to believe. In fact, the prooflooks very much like the result of trying to x the problem in the rst proof by replacing (nonunique) decimalexpansions with (unique) continued fraction expansions.March 2011] WAS CANTOR SURPRISED? 203where the partial quotients a, b, c, . . . , etc. are all positive integers. This representationis innite if and only if x is irrational, and in that case the representation is unique.So one can argue just as before, interleaving the two continued fractions for xand y, to establish a bijection between the set of pairs (x, y) such that both x and yare irrational and the set of irrational points in [0, 1]. The result is a bijection becausethe inverse mapping, splitting out two continued fraction expansions from a given one,will certainly produce two innite expansions.That being done, it remains to be shown that the set of irrational numbers between0 and 1 can be put into bijection with the interval [0, 1]. This is the hard part of theproof. Cantor proceeded as follows.First he chose an enumeration of the rationals {rk} and an increasing sequence ofirrationals {k} in [0, 1] converging to 1. He then looked at the bijection from [0, 1] to[0, 1] that is the identity on [0, 1] except for mapping rk k, k rk. This gives abijection between irrationals in [0, 1] and [0, 1] minus the sequence {k} and reducesthe problem to proving that [0, 1] can be put into bijection with [0, 1] {k}.At this point Cantor claims that it is now enough to successively apply the fol-lowing theorem:A number y that can assume all the values of the interval (0 . . . 1) with the soli-tary exception of the value 0 can be correlated one-to-one with a number x thattakes on all values of the interval (0 . . . 1) without exception.In other words, he claimed that there was a bijection between the half-open interval(0, 1] and the closed interval [0, 1], and that successive application of this fact wouldnish the proof. In the actual application he would need the intervals to be open on theright, so, as we will see, he chose a bijection that mapped 1 to itself.Cantor did not say exactly what kind of successive application he had in mind,but what he says in a later letter suggests it was this: we have the interval [0, 1] minusthe sequence of the k. We want to put back in the k, one at a time. So we leavethe interval [0, 1) alone, and look at (1, 2). Applying the lemma, we construct abijection between that and [1, 2). Then we do the same for (2, 3) and so on. Puttingtogether these bijections produces the bijection we want.Finally, it remained to prove the lemma, that is, to construct the bijection from[0, 1]to (0, 1]. Modern mathematicians would probably do this by choosing a sequence xnin (0, 1), mapping 0 to x1 and then every xn to xn+1. This Hilbert hotel idea was stillsome time in the future, however, even for Cantor. Instead, Cantor chose a bijectionthat could be represented visually, and simply drew its graph. He asked Dedekind toconsider the following peculiar curve, which we have redrawn in Figure 1 based onthe photograph reproduced in [4, p. 63].Such a picture requires some explanation, and Cantor provided it. The domainhas been divided by a geometric progression, so b = 1/2, b1 = 3/4, and so on;a = (0, 1/2), a

= (1/2, 3/4), etc. The point C is (1, 1). The points d

= (1/2, 1/2),d

= (3/4, 3/4), etc. give the corresponding subdivision of the main diagonal.The curve consists of innitely many parallel line segments ab, a

b

, a

b

andof the point c. The endpoints b, b

, b

, . . . are not regarded as belonging to thecurve.The stipulation that the segments are open at their lower endpoints means that 0 is notin the image. This proves the lemma, and therefore the proof is nished.204 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118OPCddddb1 b2 b3b4baaaa abbbbFigure 1. Cantors function from [0, 1] to (0, 1].Cantor did not even add that last comment. As soon as he had explained his curve, hemoved on to make extensive comments on the theorem and its implications. He turnson its head the objection that various mathematicians made to his question, namelythat it was obvious from geometric considerations that the number of variables isinvariant:For several years I have followed with interest the efforts that have been made,building on Gauss, Riemann, Helmholtz, and others, towards the clarication ofall questions concerning the ultimate foundations of geometry. It struck me thatall the important investigations in this eld proceed from an unproven presuppo-sition which does not appear to me self-evident, but rather to need a justication.I mean the presupposition that a -fold extended continuous manifold needs independent real coordinates for the determination of its elements, and that fora given manifold this number of coordinates can neither be increased nor de-creased.This presupposition became my view as well, and I was almost convinced ofits correctness. The only difference between my standpoint and all the otherswas that I regarded that presupposition as a theorem which stood in great needof a proof; and I rened my standpoint into a question that I presented to severalcolleagues, in particular at the Gauss Jubilee in G ottingen. The question was thefollowing:Can a continuous structure of dimensions, where > 1, be related one-to-one with a continuous structure of one dimension so that to each point of theformer there corresponds one and only one point of the latter?Most of those to whom I presented this question were extremely puzzled thatI should ask it, for it is quite self-evident that the determination of a point in anextension of dimensions always needs independent coordinates. But who-ever penetrated the sense of the question had to acknowledge that a proof wasneeded to show why the question should be answered with the self-evidentno. As I say, I myself was one of those who held it for the most likely that theMarch 2011] WAS CANTOR SURPRISED? 205question should be answered with a nountil quite recently I arrived by ratherintricate trains of thought at the conviction that the answer to that question isan unqualied yes.3Soon thereafter I found the proof which you see before youtoday.So one sees what wonderful power lies in the ordinary real and irrational num-bers, that one is able to use them to determine uniquely the elements of a -foldextended continuous manifold with a single coordinate. I will only add at oncethat their power goes yet further, in that, as will not escape you, my proof can beextended without any great increase in difculty to manifolds with an innitelygreat dimension-number, provided that their innitely-many dimensions have theform of a simple innite sequence.Now it seems to me that all philosophical or mathematical deductions that usethat erroneous presupposition are inadmissible. Rather the difference that ob-tains between structures of different dimension-number must be sought in quiteother terms than in the number of independent coordinatesthe number that washitherto held to be characteristic.5. JE LE VOIS. . . So now Dedekind had a lot to digest. The interleaving argu-ment is not problematic in this case, and the existence of a bijection between the ra-tionals and the increasing sequence k had been established in 1872. But there were atleast two sticky points in Cantors letter.First, there is the matter of what kind of successive application of the lemmaCantor had in mind. Whatever it was, it would seem to involve constructing a bijectionby putting together an innite number of functions. One can easily get in trouble.For example, here is an alternative reading of what Cantor had in mind. Insteadof applying the lemma to the interval (1, 2), we could apply it to (0, 1) to put itinto bijection with (0, 1]. So now we have put 1 back in and we have a bijectionbetween [0, 1] {1, 2, 3, . . . } and [0, 1] {2, 3, . . . }.Now repeat: use the lemma on (0, 2) to make a bijection to (0, 2]. So we have put2 back in. If we keep doing that, we presumably get a bijection from (0, 1) minusthe k to all of (0, 1).But do we? What is the image of, say, 131? It is not xed under any of our functions.To determine its image in [0, 1], we would need to compose innitely many functions,and its not clear how to do that. If we manage to do it with some kind of limitingprocess, then it is no longer clear that the overall function is a bijection.The interpretation Cantor probably intended (and later stated explicitly) yields aworkable argument because the domains of the functions are disjoint, so it is clearwhere to map any given point. But since Cantor did not indicate his argument in thisletter, one can imagine Dedekind hesitating. In any case, at this point in history theidea of constructing a function out of innitely many pieces would have been bothnew and worrying.The second sticky point was Cantors application of his theorem to underminethe foundations of geometry. This is, of course, the sort of thing one has to be carefulabout. And it is clear, from Dedekinds eventual response to Cantor, that it concernedhim.Dedekind took longer than usual to respond. Having already given one wrong proof,Cantor was anxious to hear a yes from Dedekind, and so he wrote again on June 29:3The original reads . . . bis ich vor ganz kurzer Zeit durch ziemlich verwickelte Gedankereihen zu der Ue-berzeugung gelangte, dass jene Frage ohne all Einschr ankgung zu bejahen ist. Note Cantors Uberzeugungconviction, belief, certainty.206 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Please excuse my zeal for the subject if I make so many demands upon yourkindness and patience; the communications which I lately sent you are even forme so unexpected, so new, that I can have no peace of mind until I obtain fromyou, honoured friend, a decision about their correctness. So long as you have notagreed with me, I can only say: je le vois, mais je ne le crois pas. And so I askyou to send me a postcard and let me know when you expect to have examinedthe matter, and whether I can count on an answer to my quite demanding request.So here is the phrase. The letter is, of course, in German, but the famous I see it,but I dont believe it is in French.4Seen in its context, the issue is clearly not thatCantor was nding it hard to believe his result. He was condent enough about thatto think he had rocked the foundations of the geometry of manifolds. Rather, he felta need for conrmation that his proof was correct. It was his argument that he sawbut had trouble believing. This is conrmed by the rest of the letter, in which Cantorspelled out in detail the most troublesome step, namely, how to successively applyhis lemma to construct the nal bijection.So the famous phrase does not really provide an example of a mathematician havingtrouble believing a theoremeven though he had proved it. Cantor, in fact, seems to havebeen condent [ uberzeugt!] that his theorem was true, as he himself says. He had inhand at least two arguments for it: the rst argument, using the decimal expansion,required supplementation by a proof of the Schroeder-Bernstein theorem, but Cantorwas quite sure that this would eventually be proved. The second argument was correct,he thought, but its complicated structure might have allowed something to slip by him.He knew that his theorem was a radically new and surprising resultit would cer-tainly surprise others!and thus it was necessary that the proof be as solid as possible.The earlier error had given Cantor reason to worry about the correctness of his argu-ment, leaving Cantor in need of his friends conrmation before he would trust theproof.Cantor was, in fact, in a position much like that of a student who has proposedan argument, but who knows that a proof is an argument that convinces his teacher.Though no longer a student, he knows that a proof is an argument that will convinceothers, and that in Dedekind he had the perfect person to nd an error if one werethere. So he saw, but until his friends conrmation he did not believe.6. WHAT CAME NEXT. So why did Dedekind take so long to reply? From theevidence of his next letter, dated July 2, it was not because he had difculty with theproof. His concern, rather, was Cantors challenge to the foundations of geometry.The letter opens with a sentence clearly intended to allay Cantors fears: I haveexamined your proof once more, and I have discovered no gap in it; I am quite certainthat your interesting theorem is correct, and I congratulate you on it. But Dedekinddid not accept the consequences Cantor seemed to nd:However, as I already indicated in the postcard, I should like to make a remarkthat counts against the conclusions concerning the concept of a manifold of dimensions that you append in your letter of 25 June to the communication andthe proof of the theorem. Your words make it appearmy interpretation may beincorrectas though on the basis of your theorem you wish to cast doubt on themeaning or the importance of this concept . . .4I dont know whether this is because of the rhyme vois/crois, or because of the well-known phrase voir,cest croire, or for some other reason. I do not believe the phrase was already proverbial.March 2011] WAS CANTOR SURPRISED? 207Against this, I declare (despite your theorem, or rather in consequence of re-ections that it stimulated) my conviction or my faith (I have not yet had timeeven to make an attempt at a proof) that the dimension-number of a continu-ous manifold remains its rst and most important invariant, and I must defendall previous writers on the subject . . . For all authors have clearly made thetacit, completely natural presupposition that in a new determination of the pointsof a continuous manifold by new coordinates, these coordinates should also (ingeneral) be continuous functions of the old coordinates . . .Dedekind pointed out that, in order to establish his correspondence, Cantor hadbeen compelled to admit a frightful, dizzying discontinuity in the correspondence,which dissolves everything to atoms, so that every continuously connected part of onedomain appears in its image as thoroughly decomposed and discontinuous. He thenset out a new conjecture that spawned a whole research program:. . . for the time being I believe the following theorem: If it is possible to establisha reciprocal, one-to-one, and complete correspondence between the points of acontinuous manifold A of a dimensions and the points of a continuous manifoldB of b dimensions, then this correspondence itself, if a and b are unequal, isnecessarily utterly discontinuous.In his next letter, Cantor claimed that this was indeed his point: where Riemann andothers had casually spoken of a space that requires n coordinates as if that number wasknown to be invariant, he felt that this invariance required proof. Far from wishing toturn my result against the article of faith of the theory of manifolds, I rather wish touse it to secure its theorems, he wrote. The required theorem turned out to be true,indeed, but proving it took much longer than either Cantor or Dedekind could haveguessed: it was nally proved by Brouwer in 1910. The long and convoluted story ofthat proof can be found in [3], [11], and [12].Finally, one should point out that it was only some three months later that Cantorfound what most modern mathematicians consider the obvious way to prove thatthere is a bijection between the interval minus a countable set and the whole interval.In a letter dated October 23, 1877, he took an enumeration of the rationals and let =2/2. Then he constructed a map from[0, 1] sending to 21, to 2, andevery other point h to itself, thus getting a bijection between [0, 1] and the irrationalnumbers between 0 and 1.7. MATHEMATICS AS CONVERSATION. Is the real story more interesting thanthe story of Cantors surprise? Perhaps it is, since it highlights the social dynamic thatunderlies mathematical work. It does not render the theorem any less surprising, butshifts the focus from the result itself to its proof.The record of the extended mathematical conversation between Cantor and Dede-kind reminds us of the importance of such interaction in the development of mathe-matics. A mathematical proof is, after all, a kind of challenge thrown at an idealizedopponent, a skeptical adversary that is reluctant to be convinced. Often, this adversaryis actually a colleague or collaborator, the rst reader and rst critic.A proof is not a proof until some reader, preferably a competent one, says it is. Untilthen we may see, but we should not believe.REFERENCES1. K.-R. Biermann, Dedekind, in Dictionary of Scientic Biography, C. C. Gillispie, ed., Scribners, NewYork, 19701981.208 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 1182. W. Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathe-matics, Princeton University Press, Princeton, 2007.3. J. W. Dauben, The invariance of dimension: Problems in the early development of set theory and topology,Historia Math. 2 (1975) 273288. doi:10.1016/0315-0860(75)90066-X4. , Georg Cantor: His Mathematics and Philosophy of the Innite, Princeton University Press,Princeton, 1990.5. R. Dedekind, Essays in the Theory of Numbers (trans. W. W. Beman), Dover, Mineola, NY, 1963.6. , Gesammelte Mathematische Werke, R. Fricke, E. Noether, and O. Ore, eds., Chelsea, New York,1969.7. , Stetigkeit und Irrationalzahlen, 1872, in Gesammelte Mathematische Werke, vol. 3, item L,R. Fricke, E. Noether, and O. Ore, eds., Chelsea, New York, 1969.8. W. Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford UniversityPress, Oxford, 1996.9. R. Gray, Georg Cantor and transcendental numbers, Amer. Math. Monthly 101 (1994) 819832. doi:10.2307/297512910. J. Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton UniversityPress, Princeton, 1945.11. D. M. Johnson, The problem of the invariance of dimension in the growth of modern topology I, Arch.Hist. Exact Sci. 20 (1979) 97188. doi:10.1007/BF0032762712. , The problem of the invariance of dimension in the growth of modern topology II, Arch. Hist.Exact Sci. 25 (1981) 85267. doi:10.1007/BF0211624213. H. Meschkowski. Cantor, in Dictionary of Scientic Biography, C. C. Gillispie, ed., Scribners, New York,19701981.14. E. Noether und J. Cavaill` es, Briefwechsel CantorDedekind, Hermann, Paris, 1937.FERNANDO Q. GOUVEA is Carter Professor of Mathematics at Colby College in Waterville, ME. He isthe author, with William P. Berlinghoff, of Math through the Ages: A Gentle History for Teachers and Others.This article was born when he was writing the chapter in that book called Beyond Counting. So its Billsfault.Department of Mathematics and Statistics, Colby College, Waterville, ME 04901fqgouvea@colby.eduMarch 2011] WAS CANTOR SURPRISED? 209On Legendres Work on the Law ofQuadratic ReciprocitySteven H. WeintraubAbstract. Legendre was the rst to state the law of quadratic reciprocity in the form in whichwe know it and he was able to prove it in some but not all cases, with the rst complete proofbeing given by Gauss. In this paper we trace the evolution of Legendres work on quadraticreciprocity in his four great works on number theory.As is well known, Adrien-Marie Legendre (17521833) was the rst to state the lawof quadratic reciprocity in the form in which we know it (though an equivalent resulthad earlier been conjectured by Euler), and he was able to prove it in some but notall cases, with the rst complete proof being given by Gauss [3]. In this paper wetrace the evolution of Legendres work on quadratic reciprocity in his four great works[10, 11, 12, 13] on number theory. These works span a 45 year period in Legendreslife, dating from 1785, 1797, 1808, and 1830 respectively.Before beginning with our analysis here, we call the readers attention to severalother relevant works. [15] overlaps with our work here, though it has a somewhatdifferent focus. [2] is a brief survey, and [14] is an extended treatment of the earlyhistory of reciprocity laws. A highly readable account of the development of numbertheory around this era can be found in [9], which has excerpts from original works ofEuler, Legendre, Gauss, and others, translated into English.In this paper, we will use Legendres language to the extent possible. In particular,we will not use the terms quadratic residue/nonresidue or the notion of congruence inthe body of this article, as these were never used by Legendre.We begin with Legendres 1785 paper [10]. In Article I of that paper he proves aresult due originally to Euler:Theorem A. Let c be an odd prime and let d be any integer not divisible by c. Thendc11 is divisible by c.Furthermore, c divides the formula x2+ dy2(i.e., there are integers x and y notdivisible by c with x2+dy2divisible by c) if and only if (d)(c1)/2leaves a remainderof 1 when divided by c; otherwise (d)(c1)/2leaves a remainder of 1 when dividedby c. If c/2 < d < c/2, each possibility occurs for (c 1)/2 values of d.1We follow Legendres notation throughout this paper and let a and A be distinctprimes of the form 4x + 1 and b and B be distinct primes of the form 4x 1 (or4x +3; Legendre used both but preferred 4x 1).In Article IV he rewrites the above conclusions as (d)(c1)/2= 1 or (d)(c1)/2=1, with the convention here, as he explicitly states elsewhere, that this is true omit-ting multiples of c. With this convention he then states the following 8 theorems:doi:10.4169/amer.math.monthly.118.03.2101Note that c divides the formula x2+dy2 if and only if d is a quadratic residue (mod c). To see this,rst suppose that d is a quadratic residue (mod c), and let x be an integer with x2 d (mod c). Thenx2+d12 0 (mod c). Conversely, if x2+dy2 0 (mod c), let y be an integer with yy 1 (mod c). Thenx2y2+d 0 (mod c), so d (x y)2(mod c).210 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Theorem I. If b(a1)/2= 1, then a(b1)/2= 1.Theorem II. If a(b1)/2= 1, then b(a1)/2= 1.Theorem III. If a(A1)/2= 1, then A(a1)/2= 1.Theorem IV. If a(A1)/2= 1, then A(a1)/2= 1.Theorem V. If a(b1)/2= 1, then b(a1)/2= 1.Theorem VI. If b(a1)/2= 1, then a(b1)/2= 1.Theorem VII. If b(B1)/2= 1, then B(b1)/2= 1.Theorem VIII. If b(B1)/2= 1, then B(b1)/2= 1.Note that Theorems I and II are equivalent (being contrapositives of each other) asare Theorems III and IV, and Theorems V and VI, leaving ve essentially differentcases. Legendre makes this observation in the course of his proofs.Legendre then proceeds to (attempt to) prove these theorems. As he observes, he issuccessful in unconditionally proving Theorems I, II, and VII, but for the remainingcases his proof is conditional on an auxiliary hypothesis that he cannot prove. We statethis as Hypothesis A below. His key tool in these (partial) proofs is the following resultof his from Article III:Theorem B. Let r, s, and t be squarefree, pairwise relatively prime positive integers.Then the equation r x2+ sy2= t z2has a nonzero solution in integers if and only ifthere are integers , , and such thatr2+st ,t 2sr ,t 2rsare all integers.Note in particular that in this theorem r, s, and t are not required to be prime.However, in his (partial) proofs of Theorems IVIII he uses only the cases where r, s,and t are primes or are equal to 1.Nowwe come to Legendres 1797 book [11]. In this work we see two major changesfrom his previous work.The rst change is the introduction of what we now call the Legendre symbol mn

.Here n is a prime and m is an arbitrary integer not divisible by n. From Theorem Ahe knows that m(n1)/2leaves a remainder of 1 when divided by n, and then he sets

mn

= 1 or 1 as this remainder is 1 or 1. Legendre occasionally uses this symbolwhen n = 1 as well, in which case he sets mn

= 1 for every nonzero integer m.In the opinion of the author, this is more than a notational convenience. In intro-ducing this notion, Legendre reies this concept, and makes it into an object of in-dependent study. This line of thought later led to the Jacobi symbol and the Hilbertsymbol.The second change is the introduction of the term reciprocity. We have in thiswork [11, par. (164)], a paragraph entitled:March 2011] LEGENDRES WORK ON QUADRATIC RECIPROCITY 211Th eor eme contenant une loi de r eciprocit e qui existe entre deux nombres pre-miers quelconques (Theorem containing a law of reciprocity that exists betweentwo arbitrary prime numbers).Legendre begins this paragraph by letting m and n be odd primes, and with thisimplicit assumption he states the theorem:For arbitrary prime numbers m and n, if they are not both of the form 4x 1,then nm

= mn

, and if they are both of the form 4x 1, then nm

=

mn

.These two general cases are contained in the formula

nm

= (1)m12 n12

mn

.Again, in the opinion of the author, this general heading reects not only an ap-preciation of the importance of this overall result, but a conception of it as a singleresult (rather than a collection of eight results), and a conception of it as reecting arelationship between distinct odd primes.Legendre begins his proof by observing that, in case r, s, and t are primes or areequal to 1, the conditions in Theorem B can be restated in terms of the Legendresymbol:

rst

= 1,

str

= 1,

rts

= 1.He then proceeds to develop the properties of the Legendre symbol. From his de-nition he immediately derives Na

=

Na

and Nb

=

Nb

, as (a 1)/2 is even,so (1)(a1)/2= 1, and (b 1)/2 is odd, so (1)(b1)/2= 1, and also the multi-plicativity of the Legendre symbol in general: MNc

=

Mc Nc

.2In order to prove reciprocity he must divide the proof into the same eight cases ashe did in 1785. (We warn the reader who wishes to consult the works of Legendre thathe changes the numbering of these eight cases from 1785 to 1797, leaves it alone from1797 to 1808, but changes it again from 1808 to 1830. In this paper we use the 1785numbering throughout.) His proof is essentially (with one exception, noted below) thesame as that in 1785, with some simplication due to the use of the Legendre symbol.He uses the same auxiliary hypothesis, Hypothesis A, as in 1785, which we now state(we have delayed stating it until now as it is much easier to state using the Legendresymbol).Hypothesis A.(a) For any a and A, there exists b with ba

= 1 and bA

= 1.(b) For any a and b, there exists A with Aa

= 1 and Ab

= 1.(c) For any b and B, there exists a with ab

= 1 and aB

= 1.The only essential difference between the proofs in 1785 and 1797 is that in 1797Legendre gives two proofs of cases I and II, the rst of which is the same as his 1785proof.2Nowadays it is common to dene the Legendre symbol

mn

by

mn

= 1 if m is a quadratic residue (mod n)and mn

= 1 if m is a quadratic nonresidue (mod n), but this was not Legendres denition. Of course, themodern denition and Legendres denition are equivalent, by Eulers theorem. Note that the multiplicativityof the Legendre symbol is immediate from Legendres denition, but takes some work to obtain from themodern denition.212 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Nowwe come to Legendres 1808 book [12]. First we see that Legendre has isolatedthe multiplicativity of the Legendre symbol as an important property and instead ofproving it inter alia as part of the proof of reciprocity, he proves this property in [12,par. (135)], the paragraph in which he rst introduces the Legendre symbol.The major difference, however, is that Legendre has changed his proof of reci-procity. His proof here of cases I and II is the same as his second proof of these casesin 1797. As for the remaining cases, he has a new approach, based on his earlier resultson the possibility of solving the equations Mx2 Ny2= 1 for M and N primes. Us-ing these results, he treats cases VII and VIII together, establishing them uncondition-ally. As for the remaining cases, his proof depends on a different auxiliary hypothesis,Hypothesis B, than he had previously used. His auxiliary hypothesis here is:Hypothesis B. For any a, there exists b with ab

= 1.Of course, by this time Gauss had proved the law of quadratic reciprocity in general.Gauss gave two proofs in [3] and a third in [4]. (The author admits to not being ableto read Latin, and he consulted [3, 4, 5, 6] in the German translation [7].) In [12, par.(381)], Legendre gives Gausss third proof as well.In Legendres two-volume 1830 book [13] we again nd his attempt to provequadratic reciprocity. Here his proof is the same as in 1808, with the same dependenceon Hypothesis B in some of the cases. He again gives Gausss third proof, but he alsogives a proof in [13, par. (679)] that is a modication of a proof due to Jacobi [8].Jacobis proof is in part a simplication of Gausss sixth proof [6]. Jacobi begins withwhat we now call the Gauss sum P =

c1d=1

dc

exp(2i d/c). It is easy to show thatP2= (1)(c1)/2c, so P = c if c is a prime of the form 4x +1 and P = ic ifc is a prime of the form 4x 1.It is a celebrated theorem of Gauss [5] that the sign is always positive, and in thatwork Gauss used this fact to provide his fourth proof of quadratic reciprocity. Jacobiused the value of P. Legendre modied Jacobis proof to use only the value of P2sothat the difcult sign question could be bypassed. In fact, the observation that quadraticreciprocity can be proved using only the value of P2and not the value of P goes backto Gauss in his sixth proof [6]. Legendre describes the proof he gives as the simplestof all known proofs of quadratic reciprocity.Legendre realized full well that his own two proofs of quadratic reciprocity wereincomplete. The change from his 1785/1797 proof to his 1808/1830 proof enabledhim to prove an additional case of reciprocity unconditionally. But this new proof alsoinvolved a change of the auxiliary hypothesis on which the proof of the remainingcases depended. Evidently, since he replaced Hypothesis A by Hypothesis B in hislater works, he regarded that as progress. In 1808 in [12, par. (169)], the rst timeHypothesis B appears, he observes that a, being of the form 4x + 1, is necessarilyof the form 8x + 1 or 8x + 5, and he can verify this hypothesis in the 8x + 5 case,leaving the 8x + 1 case open. He further observes that that case splits into the twocases 24x +1 and 24x +17, and he can verify this hypothesis in the 24x +17 case,choosing b = 3, leaving the 24x +1 case open. In 1830 in [13, par. (171)] he observesthat in addition he has veried this hypothesis for each of the fteen primes a of theform 24x +1 with a 1009. He veries this by considering the remainders when ais divided by 168 or 264, where he chooses b = 7 or b = 11 respectively.In fact, Legendres earlier Hypothesis A is a consequence of Dirichlets theoremon primes in an arithmetic progression, although, in the authors opinion, this is cer-tainly overkill. Ironically, there is no known proof of his Hypothesis B that does notuse quadratic reciprocity. Thus, what seemed to him to be an advance seems to usMarch 2011] LEGENDRES WORK ON QUADRATIC RECIPROCITY 213nowadays to be a step backwards. (This has been observed earlier; see for example[14, 15].)We now turn to the supplements to the law of quadratic reciprocity. The rst ofthese is that 1c

= 1 if c is a prime of the form 4x +1 and 1c

= 1 if c is a primeof the form 4x +3. This was known to Fermat and has already been remarked uponabove. The second of these is that 2c

= 1 if c is a prime of the form 8x +1 or 8x +7and 2c

= 1 if c is a prime of the form 8x +3 or 8x +5. Legendre derives this in anelementary way in 1797 in [11, par. (148)] from the following theorem:Theorem C.(a) An odd prime c is of the form y2+ z2if and only if c is of the form 8x +1 or8x +5.(b) An odd prime c is of the form y2+2z2if and only if c is of the form 8x +1 or8x +3.(c) An odd prime c is of the form y22z2if and only if c is of the form 8x +1 or8x +7.Legendre credits the discovery of all parts of this theorem to Fermat, with the rstproofs of parts (a) and (b) due to Euler and the rst proof of (c) due to Lagrange.This derivation of the value of 2c

is unchanged in 1808/1830 in [12, 13], althoughin those works Legendre investigates the equations Mx2 Ny2= 2, and some butnot all cases of 2c

follow more simply from those investigations, as he notes. (Thederivation of the value of 2c

was much simplied by Gauss. In [3] he proves this inan elementary way and in [4] he gives a second proof, using Gausss lemma, as partof his third proof of reciprocity. Both of these proofs were well known in the 19thcentury, appearing in the famous textbook [1], written by Dirichlet but with revisionsand supplements due to Dedekind, which gave (a slight reformulation of) Gausss thirdproof of reciprocity as well. But only the last of these proofs is well known today, whenit has become the standard proof of the value of 2c

.)We have concentrated here on Legendres work, but we would like to make a fewmore historical remarks. We have mentioned that Euler stated a conjecture equivalentto the law of quadratic reciprocity (though it takes a bit of work to see that), but Eulersstatement seems not to have had any inuence on either Legendre or Gauss. Eulercoined the terms quadratic residue and nonresidue, which were not used by Legendrebut were used by Gauss. Legendre coined the term quadratic reciprocity, but this wasnever used by Gauss, who always referred to this result as the fundamental theorem(in the theory of quadratic residues), nor did Gauss ever use the Legendre symbol inany of his works on the subject.In the introduction to [3] Gauss writes that his work there had been done withoutknowledge of prior results in the subject. He also writes there that in the meanwhile,the excellent work [11] of the highly deserving Legendre appeared, but that hedid not rewrite [3] to take it into account, only adding a few additional remarks in theAppendix. He makes some historical remarks in [3, par. (151)] immediately after heproves the fundamental theorem, in which he comments on the efforts of Euler andLegendre. In particular, Gauss credits Legendre for having arrived at that theorem in[10], without having been able to completely prove it (as Legendre himself admittedthere), and then claims (quite fairly, in the opinion of the author) that his own proof isthe rst proof.In the rst paragraph of [4] Gauss writes that in number theory it is often easy toinductively arrive at results whose proofs lie very deep, or even which defy proof.214 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118In the second paragraph of that work Gauss credits Legendre with being the rst todiscover the fundamental theorem, although he could not prove it, but that he him-self arrived at it independently in 1795 and was able to prove it only after a yearseffort.Legendre evidently felt that Gauss never gave himdue credit for the lawof quadraticreciprocity, nor for the discovery of the method of least squares, which they both ar-rived at as well. See [16] for a bitter commentary that Legendre wrote about Gauss.To quote excerpts from [16], in the translation given there, . . . the author [Legendre]spoke of his method of least squares: it is enough to recall that he published it for therst time, in 1805 . . . However, as a very celebrated geometer [Gauss] has not hesi-tated to appropriate this method to himself in a work printed in 1809 . . . In addition,we would have willingly spoken of the episode of 1809 as of a totally new and differ-ent kind, if we had not found that in 1801 the same geometer made another attempt ofthis type, in a way this earlier behavior was even more imperfect . . . While we mayfeel that there is enough glory in the development of number theory to go around, thatfeeling was apparently not shared by all the protagonists.Finally, one may ask why Legendre continued to include his own incomplete proofof the law of quadratic reciprocity in [12] and [13]. While this question cannot bedenitively answered without reading Legendres mind, the author speculates thatthere were two reasons. First, Legendre hoped that a (simple) proof of his auxiliaryhypothesis would be found, thus vindicating his approach, and second, to have omit-ted his proof would have been to weaken his claim of priority for this theorem, andthat is something he was certainly most unwilling to do.ACKNOWLEDGMENTS. This work was done while the author was on sabbatical leave at the MathematicsInstitute of the University of G ottingen, which he would like to thank for its hospitality. He notes with particularpleasure that the copy of [11] he consulted in G ottingen was the copy fromGausss personal library. Preparationof this manuscript for publication was supported by a Faculty Research Grant from Lehigh University.REFERENCES1. P. G. L. Dirichlet, Vorlesungen uber Zahlentheorie, Chelsea, New York, 1968; reprint of the 4th edition,Braunschweig, 1893.2. G. Frei, The reciprocity law from Euler to Eisenstein, in The Intersection of History and Mathematics,Science Networks Historical Studies, vol. 15, S. Chikara, S. Mitsuo, and J. W. Dauben, eds., Birkh auser-Verlag, Basel, 1994, 6790.3. C.-F. Gauss, Disquisitiones Arithmeticae, privately printed, Leipzig, 1801.4. , Theorematis arithmetici demonstratio nova, Commentationes soc. reg. sc. Gottingensis XVI,1808.5. , Summatio quarumdum serium singularium, Commentationes soc. reg. sc. Gottingensis recen-tiores I, 1811.6. , Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationesnovae, Commentationes soc. reg. sc. Gottingensis recentiores IV, 1818.7. , Untersuchungen uber h ohere Arithmetik (trans. H. Maser), American Mathematical Society/Chelsea, Providence 2006.8. C. G. J. Jacobi, Letter to Legendre of 5 August 1827, in Collected Works, vol. I, C. W. Borchardt, ed.,Chelsea, New York, 1969, 390396; reprint of the original edition, G. Reimer, Berlin, 1881.9. A. Knoebel, R. Laubenbacher, J. Lodder, and D. Pengelley, Mathematical Masterpieces, Further Chron-icles by the Explorers, Springer, New York, 2007.10. A. M. Le Gendre, Recherches danalyse ind etermin ee, Histoire de lAcad emie royale des sciences avecles M emoires de Math ematique et de Physique pour la m eme Ann ee, 1785, 465559; also available athttp://gallica.bnf.fr.11. A. M. Legendre, Essai sur la Th eorie des Nombres, Paris, 1797.12. , Essai sur la Th eorie des Nombres, Paris, 1808.March 2011] LEGENDRES WORK ON QUADRATIC RECIPROCITY 21513. , Th eorie des Nombres, Tomes I et II, Paris 1830. Also available in German translation in Zahlen-theorie von Adrien-Marie Legendre (trans. H. Maser), Michigan Historical Reprint Series, University ofMichigan Library.14. F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, Springer, Berlin, 2000.15. H. Pieper, Uber Legendres Versuche, das quadratische Reziprozit atsgesetz zu beweisen, Acta Hist.Leopold. 27 (1997) 223237.16. S. M. Stigler, An attack on Gauss, published by Legendre in 1820, Historia Math. 4 (1977) 3135.doi:10.1016/0315-0860(77)90032-5STEVEN H. WEINTRAUB is Professor of Mathematics at Lehigh University. His research spans a range ofareas in algebra, geometry, and topology, although lately he has become interested in the history of mathematicsas well. When he is not thinking about mathematics he can often be found reading mystery novels or ying hisairplane (but not doing both simultaneously).Department of Mathematics, Lehigh University, Bethlehem, PA 18015steve.weintraub@lehigh.eduA Rational Function Without a Rational AntiderivativeWe give a simple proof, based on the mean value theorem of calculus, that theantiderivative of 1/(1 + x2) is not rational.Assume on the contrary thatR(x) = P(x)Q(x)= p0xn+ + pnq0xm+ +qmis such thatR

(x) = 11 + x2,with p0, q0 = 0. Observe that R must be strictly increasing. Next, using the meanvalue theorem, we see that, if x > 0, there exists y between x and 2x such thatR(2x) R(x) = R

(y)x x1 + x2.HenceR(2x) R(x) = p0q0(2n2m)xn+m+ q202mx2m+ 0as x +, and so n m. Therefore R(x) approaches the same nite limit asx and x + ( p0/q0 if n = m, and 0 if n < m), contradicting thefact that it is strictly increasing.Submitted by Andr e Giroux, Universit e de Montr eal216 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Equimodular Polynomials and theTritangency Theorems of Euler, Feuerbach,and GuinandAlexander Ryba and Joseph SternAbstract. We strengthen a result of Lehmer, obtaining a new necessary condition for the rootsof a complex polynomial to have equal modulus. From this we derive the famous theorem ofFeuerbach, as well as the less well-known theorems of Euler and Guinand on the tritangentcenters of a triangle. The latter theorems constrain the possible locations of the incenter andexcenters subject to xed locations for the circumcenter and orthocenter.1. INTRODUCTION. As early as 1765, Euler knew that if a triangle is varied whileits circumcenter and orthocenter remain xed, then its incenter will stay within abounded region. Euler [3] characterized the region by a pair of simple inequalities,but did not comment on its shape.Remarkably, the shape of the region was not known until 1984, when Guinand [5]proved it to be a punctured disc. Guinands method extended nicely to the excenters,and he was able to show that each excenter ranges over the common exterior of Eulerspunctured disc and a certain simple closed curve. We refer to the boundary curves ofthese regions as the Euler shield and Guinand shield.1There are now several proofs of Eulers theorem (see, e.g., [8, 9, 10, 12, 13]). Ourapproach is motivated by [10], which derives the theorem from the fact that a certaincubic polynomial has complex roots of equal modulus. Following Lehmer, we call apolynomial equimodular if all its roots have the same nonzero modulus. In [6], Lehmergives a necessary condition for equimodularity. We strengthen this condition and applythe result to a special family of cubics, obtaining new proofs of the theorems of Eulerand Guinand. Just as Guinand explained the geometric meaning of the Euler shield, ourproof introduces a synthetic construction of the Guinand shield, shedding light on itsgeometric meaning. We are also able to extract a short proof of Feuerbachs theorem.The three theoremsEuler, Feuerbach, Guinandwe call E-F-G.2. THE THEOREMS E-F-G. Eulers 1765 article [3] marks an important milestonein triangle geometry. In it, he introduces the line now referred to as the Euler line,as well as his formula for the distance between the circumcenter and the incenter ofa triangle. These points are two of the four classical triangle centers, illustrated inFigure 1.The circumcenter O is the center of the circumscribed circle, as well as the inter-section point of the perpendicular bisectors of the sides.2The incenter I is the centerof the inscribed circle or incircle, and also the intersection point of the angle bisec-tors. The remaining classical centers are the orthocenter H, where the altitudes meet,and the centroid G, where the medians meet. In a nonequilateral triangle, the pointsO, G, and H lie on the Euler line in the order O-G-H, with GH = 2 OG. A fthtriangle center, apparently unknown to Euler, is the nine-point center N. This is thedoi:10.4169/amer.math.monthly.118.03.2171After a suggestion of John Conway.2Our notation and terminology are consistent with those of [2].March 2011] EQUIMODULAR POLYNOMIALS AND TRITANGENCY THEOREMS 217INGHOA DAB CFigure 1. The classical triangle centers and the Euler segment OH.center of the nine-point circle, which passes through the midpoints of the three sidesand the feet of the three altitudes. It turns out that N is also the midpoint of OH, sothat OG : GN : NH = 2 : 1 : 3. The nine-point circle is often attributed to Euler, butno evidence has been found to support this attribution. According to [7], there areprecedents of the nine-point circle theorem dating as far back as the 1804 work ofBenjamin Bevan, but it is rst explicitly described in the 1821 article [1] of Brianchonand Poncelet. The term nine-point circle was coined in 1842 by O. Terquem [11].Although the 1765 paper is historically important for its introduction of the Eulerline, Euler states that his primary aim is to compute the sides of a triangle in terms ofits central distances, OH, OI, and IH. He does this by showing that for a nonequilateraltriangle, the central distances determine the coefcients of a real cubic whose roots areexactly the sides a, b, c. This leads to a pair of necessary conditions on OH, OI, IH,which come fromthe fact that the cubic must have three positive real roots. Guinand [5]shows that Eulers necessary conditions are also sufcient to guarantee the existenceof a triangle with pre-specied central distances. This is now called Eulers theorem.In order to formulate Eulers theorem, we begin by dening two functionsN(O, H) = 12(O + H) and G(O, H) = 13(2O + H),where O and H are any two points. In the archetypal case when O and H are thecircumcenter and orthocenter of a triangle, N(O, H) and G(O, H) coincide with thenine-point center N and the centroid G. For brevity, we always write N = N(O, H)and G = G(O, H).Denition 1. The Euler shield determined by two points O and H, denoted by =(O, H), is the circle with GH as diameter (Figure 3).The Euler shield can also be described as the locus of a point X such that OX =2 NX. This equation denes a circle of Apollonius whose center lies on line ON.Since OG : GN : NH = 2 : 1 : 3, the values X = G and X = H satisfy the equation.As G and H lie on line ON, the circle of Apollonius has GH as a diameter. We maynow state Eulers theorem.Theorem E (Euler, 1765). Three points O, H, and I are the circumcenter, orthocen-ter, and incenter of a triangle if and only if I is inside the Euler shield and differsfrom N.218 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Another celebrated theorem of the 1765 paper is the formula3OI2= R(R 2r). (1)Here R is the circumradius, or radius of the circumscribed circle, while r is the in-radius, or radius of the incircle. Equation (1) often appears in conjunction with Feuer-bachs relationNI = 12R r, (2)which expresses the famous result that the incircle and the nine-point circle are inter-nally tangent. The term 12R represents the radius of the nine-point circle.4Both (1) and (2) have analogues in which the incenter is replaced by one of thethree excenters. These are the centers of the excircles, which touch one side of ABCinternally and the other two sides externally (see Figure 2). Each is the intersection ofone internal angle bisector with two external angle bisectors. We denote the excentersopposite A, B, C by Ea, Eb, Ec. An arbitrary excenter will be denoted by E, with being the radius of the corresponding excircle. The incircle and excircles are collec-tively called tritangent circles, and their centers tritangent centers. Theorems aboutthem are called tritangency theorems.AEaCBNC BIAFigure 2. The excenter Ea and A-excircle; Feuerbachs theorem.The analogues of formulas (1) and (2) for an excenter are as follows:OE2= R(R +2) (3)NE = 12R +. (4)Together, (2) and (4) comprise Feuerbachs theorem (illustrated in Figure 2):3Euler expresses the squared distances between the classical centers in terms of the area and the elemen-tary symmetric functions of the sides a, b, c. His expression for OI2is converted to the form (1) by substitutionof the well-known formulas 2 = r(a +b +c) and 4R = abc.4Two circles are internally (externally) tangent if and only if the distance between their centers is equal tothe difference (sum) of their radii. The nine-point circle has radius 12 R because it is the circumscribed circle ofthe midpoint triangle.March 2011] EQUIMODULAR POLYNOMIALS AND TRITANGENCY THEOREMS 219Theorem F (Feuerbach, 1822). In a nonequilateral triangle, the nine-point circle isinternally tangent to the incircle, and externally tangent to the excircles.Guinand sought constraints on the location of an excenter, just as Euler had forthe incenter. The region of possible excenters turns out to be the common exterior of and a rational algebraic curve , which we call the Guinand shield (see Figure 3).Guinands denition of this curve is stated in algebraic rather than geometric language.As we will see, turns out to have a simple geometric characterization. Like that ofthe Euler shield, it depends only on the points O and H.Denition 2. The Guinand shield determined by two points O and H, denoted by = (O, H), is dened as follows. If X is a point on the Euler shield , let X be theline HX when X = H, and the tangent line of at H when X = H. For each X ,there are two points L X such that LX = 2 OX. Each such L has a reection Pacross O. Then is the joint locus of the two points P as X varies over .The construction involved in this denition is fully illustrated in Figure 5; Figure 3shows only the resulting locus. We now state Guinands theorem.Theorem G (Guinand, 1984). Three points O, H, and E are the circumcenter, theorthocenter, and one excenter of a triangle if and only if E lies outside both and .O N G HFigure 3. The Euler shield and Guinand shield .3. INVERSIVE STABILITY. Euler and Guinand proved their theorems by gener-ating the desired triangle from the roots of real cubic equations. The roots of Eulerscubic are the sides a, b, c, while those of Guinands cubic are cos A, cos B, cos C.Continuing the approach of [10], we generate the vertices A, B, Cregarded as pointsof Cfrom a complex cubic, by squaring its three roots. The circumcenter of ABClies at 0 precisely when the associated cubic is equimodular. This observation leadsus to study the notion of equimodularity in its own right, and in this section we willdevelop a useful necessary condition for it.A polynomial f will be called inversively stable if for some r > 0, the set of com-plex roots of f is invariant under inversion in the circle |z| = r. Every equimodularpolynomial is inversively stable with respect to the circle on which its roots lie. Thefollowing lemma completely characterizes inversive stability, and therefore provides anecessary condition for equimodularity.220 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118Lemma 1. The monic polynomialf (z) = zn+cn1zn1+ +c1z +c0is inversively stable with respect to a circle of radius r > 0 if and only ifr2kck = c0 cnk, k = 0, 1, . . . , n. (5)The only possible radius for which this may occur is r = |c0|1/n.In formula (5), the bar denotes complex conjugation. Of course, cn = 1.Proof. Inversion in the circle of radius r is the map z r2/ z. This transforms f (z)into f (r2/ z), which has the same set of roots asg(z) = zn c0 f

r2 z

= zn+ c1r2 c0zn1+ c2r4 c0zn2+ + r2n c0.Since f and g are monic and have the same degree, the inversive stability of f is equiv-alent to the condition f = g. Equating coefcients and taking conjugates produces (5).When k = n, (5) becomes r2n= c0 c0 = |c0|2, or r = |c0|1/n.Taking moduli in (5) gives the following weaker condition of Lehmer [6]:rk|ck| = rnk|cnk|, k = 0, 1, . . . , n.Now let each monic polynomial be identied with its vector of coefcients. Thisputs the natural topology of Cnon the set of monic polynomials of