Thinking Geometrically Author(s): Daniel Pedoe Reviewed work(s): Source: The American Mathematical Monthly, Vol. 77, No. 7 (Aug. - Sep., 1970), pp. 711-721 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2316201 . Accessed: 10/12/2011 17:26
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http://www.jstor.org 1970] THINKING GEOMETRICALLY 711
2. S. Feferman,Arithmetization of metamathematicsin a generalsetting, Fund. Math., 49 (1960) 35-92. 3. K. G6del,What is Cantor'scontinuum problem? this MONTHLY, 54 (1947) 515-525. 4. D. W. Hall, and G. L. Spencer,II, ElementaryTopology, Wiley, New York,1955, xii+303 PP. 5. P. R. Halmos,Naive Set Theory,Van Nostrand,Princeton, N. J., 1960,vii+104 pp. 6. I. Lakatos, editor.Problems in the Philosophyof Mathematics,North-Holland, Amster- dam, 1967,xv+241 pp. 7. A. R. D. Mathias,A surveyof recentresults in set theory,Amer. Math. Soc. Proc. Pure Math., 13 (1970) AxiomaticSet Theory,to appear. 8. J. D. Monk,Introduction to Set Theory,McGraw-Hill, New York,1969, ix+193 pp. 9. R. Solovay,2fro can be anythingit oughtto be, Theoryof Models,North-Holland, Amster- dam, 1964,p. 435. 10. A. Tarski,A. Mostowski,and R. M. Robinson,Undecidable Theories, North-Holland, Amsterdam,1953, xi+98 pp.
THINKING GEOMETRICALLY* DANIEL PEDOE, Universityof Minnesota To many people the word "geometry' inevitably suggests a figure,a draw- ing. We are aware of the fact, apparently overlooked by Euclid, that we have to be very careful in arguing froma figure,that we may unwittinglyas- sume propertieswhich are not deducible fromthe given hypotheses,and may thereforearrive at incorrectlogical conclusions. This may be a partial explana- tion of the fact that the whole subject of geometry, especially elementary geometry,is under attack these days. The leader of the attack, and a very formidable person he is, seems to be my old friend Prof. Jean Dieudonn6. He is, of course,a very finegeometer, and a well-knownmember of the Bourbaki school. Dieudonn6 has made his views known on a number of occasions, and most explicitlyperhaps in a long preface to a book Linear algebraand geometry,pub- * This paperis based on lecturesgiven at the Universitiesof South Carolinaand Torontoin the springof 1968,and at MakerereCollege, Uganda and the Universityof the Witwatersrand in thesummer of 1969.The invitedaddress to the NorthCentral Section of the MAA in April, 1969contained similar material. Prof.Pedoe studiedat theUniversities of London, Cambridge, and the Institutefor Advanced Study.He heldinstructorships at Southampton,Birmingham, and London,a readershipin the Univ.of London, and Professorshipsat Khartoum,Singapore, Purdue University, and his present post,the Univ. of Minnesota. Dan Pedoe is well knownfor his elegantgeometrical expositions in manyarticles, films, and books.The latterinclude the 3 vol. Methodsof Algebraic Geometry (with W. Hodge,Cambridge U. Press,1947-1953), Circles (Pergamon Press, 1957), GentleArt of Mathematics(English U. Press, 1958,Penguin, 1963), Geometric Introduction to LinearAlgebra (Wiley, 1963), Introduction toPro- jectiveGeometry (Pergamon, 1964), and Courseof Geometry for Colleges and Universities(Cambridge U. Press,forthcoming). He receivedan MAA LesterFord Award in 1968.Editor. 712 DANIEL PEDOE [September lished in 1964, and recentlyreissued in a thirdedition and in an English trans- lation [1]. This book was writtenas a teachers' book forhigh schools in France, and introduces geometric ideas in the plane and Euclidean space via linear algebra. There are no diagrams, although theiruse is not forbiddenin this book as they are in a more advanced one, where Dieudonne talks of a "strict ad- herence to axiomatic methods, with no appeal whatsoever to 'geometric in- tuition',at least in the formalproofs: a necessitywhich we have emphasized by deliberately abstaining from introducingdiagrams in this book. My opinion is that the graduate student of today must, as soon as possible, get a thorough trainingin this abstract and axiomatic way of thinking," [2, p. 5]. The high school book begins with a set of axioms forthe real numbers,then with a set for vector spaces, and proceeds to develop linear algebra and to introduce the geometryof the Euclidean plane and space by these methods. Dieudonne says that he knows nothing about the way children between the ages of 11 and 14 think,but he thinksthat the Hilbert axioms by which Eucli- dean geometrycan be developed rigorouslyare too involved,and that a develop- ment via a simpler set of axioms is desirable. His book is written,he adds, merely to put on record for the benefitof any futurehistorian who might be interestedhow Dieudonne believes that elementarygeometry can be taught in a rational manner. It should be said, parenthetically,that using this book to teach fromis a most stimulatingintellectual adventure. In case the reader mightnot notice certainomissions in his book, Dieudonne makes it plain in his preface that he thinksFrench schools spend far too much time on special propertiesof the triangle,on trigonometry,on circles and sys- tems of circles,on conics and systemsof conics, and so on. He is very amusing when he says that books on trigonometryfilled with formulasare writtenfor astronomers,surveyors, and for writersof books on trigonometry,and that school childrenshould never be deliberatelytrained forany of these professions! He gives, in his book, the essentials of trigonometryfor use in modern math- ematics, and thereare no special formulasfor the triangle.In fact,at a meeting of the North Atlantic Treaty Organization held some years ago, Dieudonne went on recordas saying "Away with the triangle!" He does not want triangles mentioned,let alone studied in elementarygeometry at all. On another occasion Dieudonne has asked: "Who ever uses barycentric coordinates?", and in the preface to the high school book he complains that he studied systemsof circlesas a student,but has never come across them again in mathematics. If, says Dieudonne, one ever comes across a conic, one can treat it by the methods of the differentialand integral calculus, as one does other curves. In the course of this paper I shall discuss some problems-and mathematics, afterall, is concernedwith posing and solving problems-in which ideas about triangles, barycentric coordinates, systems of circles and conics help in the solution. I shall not use the differentialcalculus. Perhaps I should assure any anxious reader that there is no evidence that 1970] THINKING GEOMETRICALLY 713 too much time is spent in American high schools on the study of triangles, circles, and whatnot. I asked my sophomore students recently,working back- wards to try to discover somethingthey did know, whetheranyone knew the formulafor the area of a triangle. Eventually one student said: "One half the base times the height," at which another student remarkedvery wisely: "Oh, I only had that fora right-angledtriangle!" 1. The firstproblem I wish to discuss in which geometricalthinking is help- ful runs as follows: Perpendicularsare dropped fromthe vertices A, B, C of a triangleABC in a plane 7ronto a distinct plane 2r', formingan equilateral tri- angle A'B'C'. Show that the length of a side of A'B'C' satisfiesthe equation
3x4 - 2X2(a2 + 62 + c2) + 16A2 = 0, where a, b, c are the lengthsof the sides of the given triangleABC, and A is its area. This is a straightforwardexercise in the use of Pythagoras. It occurred to me to wonder whetherthis equation has real roots. This was not mentionedin the problem,which turnedup as an exercisein an English high school textbook. The condition is a2+b2+c2>4V\/3A; with the help of a little algebra, this con- dition is equivalent to the condition
(b2 - C2)2 + (C2 - a2)2 + (a2 - b2)2 > 0. This shows that the roots of the equation are always real, and that the inequal- ity for a triangle,which is thereforealways valid, becomes an equality if and only if the triangleABC is equilateral. This did not seem to be terriblyexciting, until I discovered that in 1919 Weitzenboeck [10] had published this inequality for a triangle,had proved it algebraically,and was obviously rather proud of his result. It had occurred to me in the meantime that a better way to show that the roots of the equation are always real is to inscribean ellipse touching the sides of the triangleABC at their respective midpoints, and to project this ellipse orthogonallyinto a circle. This is always possible, and the resulting triangle A'B'C' around the circle must be an equilateral triangle. This projection is the basis of a film I have made called OrthogonalProjection, and the problemof inscribingan ellipse in a triangle to touch the sides at their midpointsis the basis of another film called CentralSimilarities [9]. Having arrivedat an inequality forany triangleby showing that an orthog- onal projection into an equilateral triangle is always possible, I wondered whether a given triangle ABC can always be projected orthogonallyinto a triangleof givenshape, with sides ka', kb', kc'. The method,using Pythagoras,is as beforeand one ends with the equation
16A'2k4 - 2Ek2 + 16A2 = 0 whereA is the area of triangleABC, whereA' is the area of a trianglewith sides 714 DANIEL PEDOE [September
a', b', and c', and where
e = a2(-a"2 + b12+ c'2) + b2(a'2- b12+ c'2) + c2(a'2+b12 - C2) The condition forreal roots of this equation is
E2 (16>AA)2,> and thissuggested the possibilityof a two-triangleinequality: E) 2 16AA'. I wondered whetherI could show geometricallythat the orthogonalprojection is always possible, and so derive the two-triangleinequality. This is where barycentriccoordinates (areal coordinates)are useful! Taking ABC as triangle of reference,consider the inscribed conic with equation p2X2 + q2Y2 + r2Z2 - 2pqXY - 2qrYZ - 2rpZX = 0. If we consider the intersectionsof this conic with X+Y+Z=O, the line at infinity,the discriminantof the resultingquadratic equation is -pqr(p + q + r), and so we are assured that if p, q, r are all positive, that is, if the points of con- tact of the conic with the sides of ABC are all internal points, then the in- scribed conic is an ellipse. Project this ellipse into a circle by orthogonal pro- jection, and suppose that the triangleABC projects into triangle A'B'C'. The center of the ellipse projects into the center of the incirclefor triangleA'B'C'. The center of the ellipse has coordinates q + r r + p: p + q, and the centerof the circle has coordinatesa':b':c', where these are the lengths of the sides of triangleA'B'C'. Since the ratio of areas is unchangedby orthog- onal projection, and the center of the ellipse projects into the center of the incircle,we have q + r: r + p: p + q = a':b':c', so that
p q: r =- a' + b' + c': a' - b + c' : a' + bl - c'. If we choose the ratios p: q: r to satisfythese equations and project the resulting ellipse orthogonallyinto a circle, then the projection of triangleABC will be a trianglesimilar to triangleA'B'C'. We have not yet finishedwith the geometricalideas associated with this two- triangleinequality. In 1937 Finsler and Hadwiger pursued the ideas of Weit- zenboeck, and derived his triangle inequality as follows: [3]. They erected equilateral triangles A"BC, B"CA and C"AB inwards on the sides BC, CA 1970] THINKING GEOMETRICALLY 715
and AB respectivelyof triangleABC. Let V be the centroid of B"CA and W the centroidof C"AB. Then by trigonometry,
(VW)2 = k(a2 + b2+ c2 - 4V3A), where k is a positive constant for the triangleABC. Since (VW)2>0, we have a2+b2+c2_ 4A/3A,with equality if and only if the triangleABC is equilateral. I was certain that this method would give my two-triangleinequality. I erected trianglesA "B C, B"CA, and C"A B inwardson the sides B C, CA and AB respectivelyof triangleABC, each similarto a triangleA 'B'C', and then I had to determinewhat would correspondto the centroidof an equilateral trianglefor these triangles A"BC, B"CA and C"AB. Anyone who has worked with tri- angles knows that the centroid has no useful angles associated with it, but the circumcenterof a trianglehas, and I found that the square of the distance be- tween the circumcenterV of B"CA and the circumcenterW of C"AB is given by the formula
2(VW/a')2 = (R'/cab'c')2[E a2(-a'2 +b 12 + c'2) -16AA'], where R' is the circumradiusof triangleA'B'C'. This shows that the triangle UVWis similarto the triangleA'B'C', and also gives the two-triangleinequality
a2( -a'2 + b'2 + c'2) + b2(a'2 - b'2 + c'2) + c2(a'2 + b12- c'2) 16MA', with equality if and only if the two trianglesABC, A'B'C' are similar. This furthercondition comes immediatelyfrom the construction.I imagine that if Finsler and Hadwiger had thoughtof this,they would have been ratherexcited, since they spent so much time on provingthe inequality forone triangle.If one of the two trianglesis already equilateral, the two-triangleinequality simplifies to the triangleinequality forthe other triangle.There is a much easier method forobtaining it [5], of course. One final remark on this discovery. Since my note on the two-trianglein- equality mentionedorthogonal projection, it was rejected by a journal I sent it to, as being "unsuitable." I rewrotethe note, and gave the derivationfrom the method of Finsler and Hadwiger, illustrious names, after all, and obtained publication in a "Research Note" [6]. This was in 1943, and things published then did not travel, but I think that this discovery of the firstinteresting in- equality fortwo trianglesdoes show how geometricalthinking, necessarily based on some knowledge of geometry,can be fruitful. 2. My second look at geometricalthinking is bound up with what is called the Six-Circle Theorem. We have four circles Cl, C2, C3, C4 in the inversive plane, and we suppose that C1, C2intersect in the points P1, Ql, that C2,C3 inter- sect in P2, Q2, that C3, C4 intersectin P8, Q3,and that C4, Ci intersectin P4, Q4. The theoremsays that if the fourpoints Pi, P2, P3, P4 lie on a circle,then so do the four points Ql, Q2, Q3, Q4. (Figure 1.) This is a well-knowntheorem, and can be proved in a numberof ways. It is 716 DANIEL PEDOE [September
FIG. 1
FIG. 2 an excitingtheorem, and I am not just representingmy own state of mind about it. It makes even the Russian writerJaglom excited. He gives it in his recent rather curious book Complex numbersin geometry[4, p. 35], where he says: 1970] THINKING GEOMETRICALLY 717
"This proposition seems rather elegant, but not particularly promising-an ordinary theoremof which there are many in elementarygeometry. However, the consequences which follow from this simple theorem can surely be called remarkable. As the firstof these consequences we mention a whole series of theoremsdue to the English geometer Clifford."One chain of theoremsbegins with threelines, which determinethree points by theirintersections. There is a unique circle throughthese points. Four lines determinethree sets of threelines, and thereforedetermine four circles. As a consequence of the Six-CircleTheorem these four circles meet in a point, which Jaglom calls the central point of the fourlines. Five lines determinefive sets of fourlines, and each tetrad determines a central point. These five central points can be shown to lie on a circle,which Jaglomcalls the centralcircle of the fivelines, and so on. We are in the inversive plane, and must accept the fact that all lines contain the point at infinity,so that in the Six-Circle Theorem any one of the circles may be a line, and among the intersectionsof two lines there is always the point at infinity. I agree with Jaglom that complex numbersafford a ready means of proving theoremsin plane geometry,but in his book he goes on to develop the theoryof dual complex numbers at great length. These are numbers of the forma+eb, where a, b are real and e2l 0. The application is to oriented circles, that is to circleswith a sense, and he obtains, aftermany pages of work,another attractive theorem,on orientedcircles and theirtangents. If lp,C2, C3,C4 are fouroriented circles, and the tangents to C1, C2 are pi, qi, the tangents to C2, C3 are P2, q2, the tangentsto C3, C4 are p3,q3, and the tangentsto C4, C1 are P4, q4, thenif thereis an orientedcircle which touches the orientedlines Pi, P2, P3,P4, thereis also an oriented circle which touches the oriented lines ql, q2, q3, q4. (Fig. 2.) I shall show that both the Six-Circle Theorem and the theoremon oriented circlesarise quite naturallyif we map the circlesof the Euclidean plane onto the points of E3. We map the circle with equation X2+ Y2- 2pX- 2qY+r= 0 onto the point (p, q, r) of E3. This is a simple, yet fundamentalmapping (it is equivalent to stereographic projection), and it does illustrate what Henri Lebesgue has referredto as one of the importantaspects of geometry,that it is almost a plastic art. Seeing theoremsfrom different aspects, I believe, is one of the great charms of geometry.In the mapping I have just described,those sys- tems of circles called coaxial systems,or pencils of circles(given by the equation kC+k'C' =0, where C and C' are given circles, and k, k' vary over the real numbers)are mapped onto the lines of E3. Is there anythingmore natural than a line in E3? There appears a quadric Q of equation X2+ Y2-Z=0, and the points of this quadric representthe point-circlesof the plane, those with zero radius. We soon find ourselves considering theorems which are obviously theoremsof projective space, and there is a most attractive interplaybetween theoremsfor circles in the plane and theoremson points and lines in projective space, S3. All thisis workedout in [7], and in moredetail in [8]. 718 DANIEL PEDOE [September
The Six-CircleTheorem becomes the followingtheorem in'S3. Suppose A, B, C, D are four distinct points in S3, suppose Q is a given quadric, and suppose AB intersectsQ in Pi, Qi. Assume also that the line B C intersectsQ in the points P2, Q2, the line CD intersectsQ in the points P3, Q3, and finallythe line DA inter- sectsQ in thepoints P4, Q4. Then ifthe points P1, P2, P3, P4 lie in a plane,so do the points Ql, Q2, Q3, Q4. This theoremcan be proved by using one of the most remarkabletheorems on quadric surfacesin S3, the theoremof the eight associated points. This says that if three quadrics intersectin eight distinctpoints, then any quadric which is made to pass throughseven of these points will automatically pass through the last point of the eight. The eight points Pi, Qi(i = 1, 2, 3, 4) are associated, since we can findthree distinct quadrics which contain them. One is the quadric Q itself,another is the reducible quadric which consists of the planes ABD and BCD, and a thirdis the reducible quadric which consistsof the planes ABC and ADC. The two reducible quadrics intersectin the lines AB, BC, CD, and DA, so the threequadrics have only the eight points Pi, Qi in common. Let the plane in which the four points Pi lie be denoted by ir, and let the plane Q1Q2Q3 be denoted by w'. Then the reducible quadric formedby the planes 'r and ir' con- tains seven of the eight associated points,and must thereforecontain the eighth point Q4. If thisdoes not lie in ir, it must lie in ir'. But it is seen immediatelythat if Q4 iS coplanar with the points P1, P2, P3, P4, then the points A, B, C, D are coplanar, and the theoremis trivial. Of course, since we are using the theoremof the eight associated points in all its strength,we must be certain that our proofof this theoremholds forall possible cases. Unfortunately,few textbooks give an acceptable proof. In S3 we have a Principle of Duality, and so our theorem has an evident dual. When we write this down, and consider how orientedcircles can be repre- sented in E3, this theoremfor S3, interpretedin E3, gives us the theoremwhich Jaglom only proves after many pages of work on dual complex numbers. I shall not give the details here. They can be found in [8].
3. My last example on thinkinggeometrically is about conics, and I shall leave the question open as to whetherthe methods of the calculus would give a complete solution of the followingproblem. Given five distinct points in the Euclidean plane, we can draw a unique conic throughthem. If we begin with six distinct points, there are six conics which can be drawn to pass respectively throughsets of fiveof the six given points. The problemis: can the six points be chosen so that (a) the six conics are all distinctellipses, or (b) the six conics are all distinct hyperbolas, or (c) the six conics are all distinct parabolas? This question was asked some years ago by C. V. Durell, senior mathematicsmaster at Winchester,one of Britain's great public (private) schools. The solution was given at sight by Dr. Beniamino Segre when I showed him the problem. Dr. Segre, who was then in England, is an illustrioussuccessor of the great Italian geometersCorrado Segre, Guido Castelnuovo and Francesco Severi. The Segre 1970] THINKING GEOMETRICALLY 719 theoremis that it is possible to findn> 5 points in the Euclidean plane so that the (5) conics throughsets of five of them are either all distinctellipses or dis- tincthyperbolas, but it is impossibleto findsix points so that the conics through sets of five of them are all distinctand parabolas. The proofsof the firsttwo statementsare similar,but the proofof the third part of the theoremhas a completelydifferent flavor, as we shall see. If we con- sider fivepoints (xi, yi) (i= 1, 2, 3, 4, 5), we can obtain the equation of the conic which passes throughthem in various ways, by writingdown a determinant, for instance. We are interestedonly in the highest degree terms. If these are AX2+2HXY+B Y2. then the conic is an ellipse if and only if H 2-AB
If we are skilled in the use of birational mappings of the projective plane (Cremona transformations),we can map the sextuplet of lines on the surface representedby the conics throughsets of fivepoints of the plane onto six points of the plane, and see what happens. The space cubic curve, originallymapped onto a line of the plane, becomes an irreduciblecurve of order fivewith cusps at the six fundamental points which now represent the parabolas through the originalsets of fivepoints. The orderis five,because cubics throughthe six new points still representplane sections of the surface,and if 3N- 6 =23, we have N=5. The singularitiesmust be cusps because the space cubic touches the lines which are now mapped onto points of the plane. But there is no irreduciblecurve of order five with cusps at six distinct points. If we apply Pluecker's Formula to obtain the number of variable tan- gents which can be drawn to this quintic curve froman arbitrarypoint of the plane, we obtain the number 5.4-6.3=2. It can be proved, however,that the only irreduciblecurve to which two variable tangents can be drawn froman arbitrary point in the plane is a conic. We thereforehave a contradiction. 4. Not very long ago a mathematician,hot in the pursuit of modern math- ematics, complained to me that it is not easy to find a book which deals with the plane representationof the cubic surface,and he needed it for his work. I met a young mathematicianin Oxford recentlywho has studied the new alge- braic geometry,but feels he must now learn some geometry. And there are otherswho have the new algebraic techniques at theirfinger tips but are search- ing for geometric ideas to motivate their research. There is no doubt that geometryshould not be neglected,although the feelingagainst it in some quar- ters is almost Freudian in its intensity.The grudges arise from having been taught too much geometry,or having been taught geometryvery badly. There is no reason why a course in geometryshould not be well taught,and the danger of having too much is no longer a serious one. The United States is passing through a transitionperiod in the teaching of geometry,and it should soon catch up with other countries,such as the Soviet Union, which have always given a lot of attention to the teaching of geometry.If one could findnothing else to say about geometry,it has always been conceded that students enjoy it, and the Pleasure Principle should not be neglected in mathematics. I have tried to communicate this feelingin my recent book [8], in which I have put down the geometricideas which turn up in mathematics.
References 1. Dieudonn6,Linear Algebra and Geometry,Hermann, Paris, 1964. 2. , Foundationsof ModernAnalysis, Academic Press, New York, 1960. 3. Finslerand Hadwiger,Comment. Math. Helv., 10 (1937) 316-26. 4. Jaglom,Complex Numbers in Geometry,Academic Press, New York,1968. 5. Pedoe,Problem E 1562,this MONTHLY, 70 (1963) 1012. 6. , Proc. CambridgePhilos. Soc., 38 (1943) 397-98. 7. , Circles,Pergamon Press, London, 1957. 1970] WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION 721
8. Daniel Pedoe,A Courseof Geometry for Colleges and Universities,Cambridge University Press,England (in the press). 9. Universityof MinnesotaCollege Geometry Project. 10. Weitzenboeck,Math. Zeit.,5 (1919) 137-146.
THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION J. H. McKAY, OaklandUniversity The followingresults of the thirtiethWilliam Lowell Putnam Mathematical Competition held on December 6, 1969 have been determinedin accordance with the regulationsgoverning the Competition.This competitionis supported by the William Lowell Putnam Intercollegiate Memorial Fund left by Mrs. Putnam in memoryof her husband and is held under the auspices of the Math- ematical Association of America. The first prize, five hundred dollars, is awarded to the Department of Mathematics of Massachusetts Instituteof Technology,Cambridge, Massachu- setts. The members of the team were Don Coppersmith,John J. Keary, and JeffreyC. Lagarias; to each of these a prize of one hundred dollars is awarded. The second prize, four hundred dollars, is awarded to the Department of Mathematics of Rice University,Houston, Texas. The members of the team were Alan R. Beale, David A. Cox, and James B. Hobelman; to each of these a prize of seventy-fivedollars is awarded. The third prize, three hundred dollars, is awarded to the Department of Mathematics of the Universityof Chicago, Chicago, Illinois. The membersof the team were Robert B. Israel, David S. Fried, and Kiyoshi Igusa; to each of these a prize of fiftydollars is awarded. The fourthprize, two hundred dollars, is awarded to the Department of Mathematics of Harvard University,Cambridge, Massachusetts. The members of the team were George Sicherman,Gerald I. Myerson,and Mark A. Mostow; to each of these a prize of fiftydollars is awarded. The fifthprize, one hundred dollars, is awarded to the Department of Mathematics of Yale University,New Haven, Connecticut.The membersof the team were Gregg J. Zuckerman,J. Lance W. Jayne, and Frederic B. Weissler; to each of these a prize of fiftydollars is awarded. The five personsranking highest in the examination,named in alphabetical order, are Alan R. Beale, Rice University; Don Coppersmith,Massachusetts Institute of Technology; Gerald A. Edgar, University of California at Santa Barbara; Robert A. Oliver, University of Chicago; Steven Winkler, Massa- chusetts Institute of Technology. Each of these has been designated as Putnam Fellows by the Mathematical Association of America and is awarded a prize of two hundred and fiftydollars. The five persons ranking second highest in the examination, named in