Mathematical Reminiscences

Mathematical Reminiscences

Howard Eves
Series: Spectrum
Copyright Date: 2001
Edition: 1
Pages: 197
https://www.jstor.org/stable/10.4169/j.ctt13x0n2s
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  • Book Info
    Mathematical Reminiscences
    Book Description:

    Long known as a mathematical storyteller, Howard Eves here writes his personal reminiscences—mostly mathematical, some not. The cast of characters includes Albert Einstein, Norbert Wiener, Julian Lowell Coolidge, Maurice Fréchet, Nathan Altschiller-Court, G. H. Hardy, and many other interesting figures whom he encountered in a long and active life in mathematics.

    eISBN: 978-0-88385-965-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Foreword
    (pp. vii-viii)

    I suppose it is natural late in one’s life to look back and enjoy again some of the more precious moments of earlier life. A number of my friends have urged me to record a collection of these earlier moments. It didn’t take much reflection to realize that, if I were to keep the treatment within reasonable bounds, I must limit myself by some rule of selection. I decided to concentrate chiefly on those moments more or less connected either with mathematics or with teaching, two pursuits that have occupied such a large part of my life.

    And so, with...

  3. Acknowledgements
    (pp. ix-x)
  4. Table of Contents
    (pp. xi-xii)
  5. The Mystery of the Four-Leaf Clovers
    (pp. 1-8)

    It was in 1955 that the Hafner Publishing Company of New York City brought out G. W. Dunnington’s scholarlyCarl Friedrich Gauss, Titan of Science: A Study of His Life and Work. I was teaching at the University of Maine at the time and, as I was then in the process of building up a history of mathematics library, I ordered a copy of the book. I keenly looked forward to the book’s arrival, and was delighted a couple of weeks later to find it on my office desk, amongst an assortment of other mail, delivered there for me while...

  6. A Fugue
    (pp. 9-14)

    (A fugue is a musical composition in which different parts are serially introduced, following one another in the manner of a flight or chase. The name derives from fuga, the Latin word for flight. The main parts of a strictfugue are an exposition, conducted by four voices, followed by a discussion. The exposition begins with the subject (theme) in the first voice. The second voice gives the answer, which is merely the subject transformed a fifth upward or downward. The third voice then introduces the original subject an octave below or above the principal key, and the fourth voice paralleling...

  7. Tombstone Inscriptions
    (pp. 15-16)

    A group of scholars, composed of a mathematician, a physicist, a chemist, a biologist, a novelist, and a school teacher, were assembled at a lunch table when the mathematician said, “Suppose, like Archimedes’ request that the geometrical figure that led him to the discovery of the formulas for the area and volume of a sphere be engraved on his tombstone, what might each of us wish to have inscribed on our tombstones? I think I would like the figure of my simple ‘proof without words’ of a complicated trigonometric identity to be inscribed on my tombstone.” The physicist said he...

  8. The Two Lights
    (pp. 17-18)

    It was around 300 BC that two lights were simultaneously lit in ancient Alexandria.

    One of these lights was a physical light in the form of the world’s first lighthouse, designed to guide ships coming down the Mediterranean Sea into the Great Harbor of Alexandria. Prior to this there had been large bonfires on shore to help guide navigators, but this was the first lighthouse in the modern sense of the word. It was built on the eastern end of a long island that lies closely off the coast of Egypt. The island was named Pharos Island, and so the...

  9. MMM
    (pp. 19-28)

    In a later reminiscence I will tell how an episode narrated by Florian Cajori in hisA History of Mathematicsled to a triple subscription to The Scholar’s Creed. In the present reminiscence I shall tell how another episode narrated by Cajori in his book led to the creation of a remarkable collection entitled MMM. Let me explain.

    In his book Cajori comments on Napoleon Bonaparte’s interest in and flair for geometry. There is now a theorem and a construction problem each bearing Napoleon’s name. Napoleon did not originate either the theorem or the construction problem; it was his popularization...

  10. Acquiring Some of the Personal Items for MMM
    (pp. 29-34)

    I mentioned, in the tale devoted to MMM, some of the personal items that ended up in that collection. It may be that certain readers are interested in how those items were secured. Here is the story of a few of them.

    When I was a graduate student at Harvard University, a friend and I sought a ride to New Haven to attend a mathematics meeting held at Yale University. In those days very few students possessed cars, so we went about contacting professors and finally secured transportation in the car driven by Professor Norbert Wiener of MIT. Professor Wiener...

  11. Difficulty in Explaining Relativity Theory in a Few Words
    (pp. 35-36)

    Reminiscing is often like a chain reaction. One reminiscence will recall another, and then that other will recall a still further one, and so on.

    The mention of Mrs. Veblen in the previous reminiscence recalled to my mind the weekly afternoon teas she used to give, at her home in Princeton, for the local mathematicians and their friends. These were very pleasant affairs, and gave the mathematicians an opportunity to talk a little “shop” among themselves.

    Mrs. Veblen dearly wished that at some time Dr. Einstein would attend one of her teas, but Dr. Einstein was not keen about such...

  12. Difficulty in Obtaining a Cup of Hot Tea
    (pp. 37-38)

    Continuing the chain reaction set up in the previous two reminiscences leads me to the following little story. Again it concerns G. H. Hardy.

    At the top of Fine Hall (Princeton’s mathematics building) was the university’s great mathematics library. Around the outer edges of the library were a number of dormer nooks, each equipped with a chair and a desk. These nooks were for the use of the graduate mathematics students and visiting mathematics scholars. I was assigned one of these nooks and Professor Hardy was given one.

    Now every afternoon, at four o’clock, there would be a soft tinkle...

  13. Hail to Thee, Blithe Spirit!
    (pp. 39-46)

    The great dream of every teacher of mathematics is to find among his or her students a potential future mathematician, and to play a beneficial part in nourishing the mathematical growth of that student. There cannot be many joys that exceed such an experience. I have been very lucky in this aspect of teaching and I could easily draw up quite a list of former students who have given me this supreme joy. I doubt they realize how deeply I thank them for having been students of mine. I have selected the following reminiscence to illustrate this aspect of teaching,...

  14. C.D.
    (pp. 47-50)

    No, the C. D. does not stand for “certificate of deposit,” but for a dear friend of mine. Working, as I have, for such a long period of time with other mathematicians, I have acquired many mathematical friends. I could devote quite some space to telling about them. Prominent in any such account would be Clayton Dodge, now a University of Maine Emeritus Professor of Mathematics.

    I first met Clayton shortly after I joined the mathematics staff at the University of Maine back in 1954. At that time Clayton was a sharp, keenly interested, and outstanding graduate mathematics student. I...

  15. Cupid’s Problem
    (pp. 51-52)

    For 25 years I had the enormous pleasure of serving as Editor of the Elementary Problems Department ofThe American Mathematical Monthly, succeeding Professor H. S. M. Coxeter of the University of Toronto when he retired from that position. One of the enjoyable features of the position lay in the resulting interesting correspondence with a large number of both domestic and foreign mathematicians. I got to know a great many of these mathematicians and the work that currently engaged them. But in time, with the huge growth of the membership of the MAA, it became essentially impossible for a single...

  16. The Lighter Life of an Editor
    (pp. 53-58)

    Many amusing things occur over a long tenure of editorship, at least so it happened to me during my more than twenty-five years as editor of the Elementary Problems Department ofThe American Mathematical Monthly. Here are three instances of this lighter side of editorial work.

    The Thompson case came about in this way. On occasion I found in my editorial work that I was able to discover a considerably better solution to a problem than any of those that were submitted. Indeed, sometimes my solution was the only one I possessed. Feeling it uncomfortably forward to publish my own...

  17. The Two Kellys
    (pp. 59-62)

    There are two Kellys, Leroy Kelly and Paul Kelly, each of whom has graced the American field of mathematics and, in particular, the area of geometry. Naturally, as geometers, I had corresponded with each Kelly for a number of years, but for a long time had not had the pleasure of meeting either one of them. The story of how I finally met these two mathematicians is interesting.

    One summer in the 1960s I was teaching at the University of Maine in a National Science Foundation Summer Institute for Teachers of Mathematics. Driving home from class one day, as I...

  18. Some Debts
    (pp. 63-64)

    I’ve lectured in every state of the union except Alaska, and many times in several states. Another frequent lecturer is Father Stanley J. Bezuszka, of Boston College. Father Bezuszka is a little human dynamo, and it was natural that we would occasionally appear on the same speaking program. It was in this way that we got to know one another.

    Thus, during an intermission of an NCTM meeting held in Corpus Christi some years ago, I found myself at a little table enjoying a Coke with Father Bezuszka. The conversation turned to geometry and the old master, Euclid. I finally...

  19. Hypnotic Powers
    (pp. 65-66)

    In 1943, while teaching Mechanics at Syracuse University, I discovered that I possess hypnotic powers. In the class was a lovable, but mentally poorly equipped, student who went, among his classmates, by the simple name of Snuffy. One day, early in the semester, I tried to make clear the difference between conclusions arrived at by deduction and those arrived at by induction—the former (if the premises are accepted) being incontestable, while the latter are only more or less probable. I illustrated deductive reasoning by such examples as:

    Premises:

    (1) All Canadians are North Americans.

    (2) All Nova Scotians are...

  20. Founding the Echols Mathematics Club
    (pp. 67-70)

    It was, if I recall correctly, during my junior year (1933) at the University of Virginia that a group of us mathematics students decided we would like to establish a university mathematics club. It wasn’t long before a charter was drawn up and an appropriate schedule of times of meeting decided upon. There only remained the matter of naming the club. I suggested we call it The Sylvester Mathematics Club, basing my suggestion on the following bit of early history of the university.

    In 1841 J. J. Sylvester accepted an appointment of Professor of Mathematics at the University of Virginia....

  21. Meeting Maurice Fréchet
    (pp. 71-74)

    It was in the mid-1940s that the great French mathematician Professor Maurice Fréchet toured up the west coast of the United States, visiting a number of institutions of higher learning on the way. When he reached Corvallis in Oregon, we members of the Mathematics Department at Oregon State College*had the pleasant opportunity of meeting the celebrated man. Professor Milne, the fatherly chairman of our department, hosted the event at his attractive home in the scenic countryside a couple of miles from Corvallis.

    It was Professor Fréchet who, in 1906, inaugurated the study of abstract spaces, and very general geometries...

  22. Mathematizing the New Mathematics Building
    (pp. 75-80)

    When I first arrived on the University of Maine campus, the Mathematics Department Office was housed in Stevens Hall along with a few mathematics classrooms, and offices and classrooms of other disciplines. The rest of the mathematics classrooms were scattered in various buildings around the campus, some even in a temporary wooden structure.

    Then a large, new, four-story building was constructed, to be shared by the English and Mathematics Departments. The building ran roughly north and south, with foyers and two auditoriums at the north end. The rest of the building was devoted to classrooms and offices, with the northern...

  23. Finding Some Lost Property Corners
    (pp. 81-84)

    When I began seeking a teaching position the great depression was at its worst, and openings in teaching were almost nonexistent. To mark time until the situation might better itself I found a job assisting a surveyor. The following year I took examinations and became a licensed land surveyor for the state of New Jersey. Rather than compete with the established city surveyors, I set up practice in the country and performed farm and vacation plot surveys. Although the remuneration was slight, it was a healthful and enjoyable outdoor activity. There was one farm survey I did for a hundred...

  24. The Tennessee Valley Authority
    (pp. 85-88)

    In 1941 the Tennessee Valley Authority in Chattanooga hired me as one of a group of mathematicians to convert logarithmic surveying forms to calculating machine surveying forms. It was the heyday of the calculating machines—the Marchant, the Frieden, and the Monroe. These machines were about the size and heft of the then standard typewriters. TVA chose to use Monroe machines.

    The forms had to be arranged so that once the given information was inserted, any person familiar with a calculating machine could step by step arrive at the desired final result. That is, the forms had to be self-explanatory....

  25. How I First Met Dr. Einstein
    (pp. 89-90)

    I hadn’t been on the Princeton campus but a few days when, one morning, while walking along a street of the village, I saw Dr. Einstein plodding along in the opposite direction on the other side of the street. I had forgotten that this was the year he was to assume his life-time appointment at the Institute for Advanced Study—that he and I both would be “freshmen” together on the Princeton campus. I paused to watch the great scientist pass by. While so engaged I noticed a young fellow on the other side of the street walking toward Dr....

  26. Catching Vibes, and Kindred Matters
    (pp. 91-92)

    One winter day at Princeton, after a light snowfall, I accompanied Dr. Einstein over to Fine Hall. As we were walking along, I sensed that we were being followed, so I turned my head and looked back. There, about a dozen paces behind us, I saw a freshman physics student, whom I knew, carefully putting his feet one after the other in Dr. Einstein’s footprints. He did this for about half of a block. The next day I met the student and asked him why the day before he had walked in Dr. Einstein’s footprints.

    “Oh,” he said. “I had...

  27. A Pair of Unusual Walking Sticks
    (pp. 93-94)

    One morning at Princeton, as I was walking over to Dr. Einstein’s home to pick him up for an early morning ramble, I passed a curbside pile of brush awaiting the town’s disposal truck. The brush consisted largely of curved canes about four feet long, an inch thick at one end and tapering to about a quarter of an inch thick at the other end. I selected one of the canes and found that if I held the thick end in my hand, extended the cane (arched upward) down to the sidewalk in front of me, and walked forward, the...

  28. A New Definition
    (pp. 95-96)

    I have told over sixty Einstein anecdotes in my various Circle books, but only those in which I was personally involved can properly be considered as reminiscences. Here are three that, when I told them in my Circle Books, I felt too forward to mention my small role in them. But since they all evoke pleasant personal memories I will here retell them in more complete form. The first two illustrate Dr. Einstein’s sense of humor.

    One day I accompanied Dr. Einstein to a lecture by a visiting physicist. The lecture was extremely dull, and the speaker droned on and...

  29. Dr. Einstein’s First Public Address at Princeton
    (pp. 97-100)

    There was a small auditorium, that I estimated would hold about 200 people, located in the center of the ground floor of Fine Hall (Princeton University’s mathematics building). It was furnished with comfortable fold-up theater seats and a small stage with a blackboard. I was informed that the auditorium was used by occasional visiting scholars to give lectures to interested brethren. It seemed to me, with the Institute for Advanced Study based at the University (the Institute did not have its own buildings at that time), the little auditorium would be an ideal place for regular bi-weekly meetings at which...

  30. Parting Advice
    (pp. 101-104)

    As narrated elsewhere in these notes, after leaving Princeton University I marked time as a surveyor while waiting for a teaching position to open up for me. I finally secured a one-year appointment at Bethany College in West Virginia, filling in while the regular man was on leave working on his Ph.D. at the University of Chicago.

    In the fall of 1938, before leaving for West Virginia, I decided to make a short nostalgic last visit to Princeton, and while there perhaps briefly to see Dr. Einstein again. I was lucky, for I found the scientist at home. He greeted...

  31. Two Newspaper Items and a Phone Call
    (pp. 105-108)

    At one time I lived directly across the street from a church officiated over by a fundamentalist minister who became concerned about my life in the hereafter. He told me grim stories about the lower regions and described the torrid horrors I might expect if I didn’t snap to and change my ways by joining his church. One regretful day, in a jest that he completely failed to appreciate, I told him that I had no fear of the lower regions, because all the mathematicians and engineers who have gone there have undoubtedly remarkably improved the place with air conditioning...

  32. Wherein the Author is Beasted
    (pp. 109-110)

    In one of my history of mathematics classes at the University of Maine I lectured ongematriaorarithmology. I explained that since many of the ancient numeral systems were alphabetical systems, it became natural to substitute number values for the letters in a name. It was this that led to the mystic pseudo-science known as gematria, or arithmology, which became very popular among the ancient Hebrews and others, and was later revived during the Middle Ages. Part of this later gematria was the art of beasting—that is, cunningly pinning onto a disliked individual the hateful number 666 of...

  33. The Scholar’s Creed
    (pp. 111-118)

    There were three of us in our sophomore year at the Eastside High School of Paterson, New Jersey, who got to be known as thethree math nuts. We were Al (Albert), Lou (Louis), and myself. We lived and breathed mathematics and were never happier than when we were struggling over some difficult mathematics problem.

    We had been hearing a great deal about a remarkable branch of mathematics called “the calculus.” Though today it is not uncommon to find a beginning course in calculus taught in high school, back then the practice was unheard of, and one had to await...

  34. The Perfect Game of Solitaire
    (pp. 119-124)

    A list of some of the principal requirements for a good game of solitaire would surely include:

    I. The rules of the game should be few and simple.

    II. The game should not require highly specialized equipment, so that it can be played almost anywhere and at almost any time.

    III. It should be truly challenging.

    IV. It should possess a number of interesting variations.

    Judged by the above requirements, the Greeks of over 2000 years ago devised what can perhaps be considered a perfect game of solitaire—it might now be called thegame of Euclidean constructions.

    The rules...

  35. The Most Seductive Book Ever Written
    (pp. 125-128)

    Occasionally, when a group of people get together enjoying one another’s company, someone will bring up the old hypothetical situation of a lone castaway stranded on a desert island. The island is “friendly,” in the sense that it possesses a mild equitable climate and a plentiful supply of easily obtained food. Since the island is far from all sea lanes and air routes, the stranded individual knows he will remain there alone, unrescued, for a considerable time, perhaps years. Now if this individual can wish to have any book he desires washed ashore in a waterproof container, what book should...

  36. The Master Geometer
    (pp. 129-130)

    During my second year in the Graduate School at Harvard University I had both the honor and pleasure of working as a geometry research student under the great geometer Julian Lowell Coolidge, who, at that time, was probably the foremost geometer on the American continent. In addition to having a head crammed with an incredible stock of geometrical knowledge, Professor Coolidge possessed a charming wit and sense of humor. He once remarked to me, “When I teach I try to make my students laugh, and when their mouths are open, I put something in for them to chew on.”

    Julian...

  37. Sandy
    (pp. 131-132)

    A relaxed atmosphere pervaded the monthly meetings of the Harvard Graduate Mathematics Club. Professor Coolidge often brought with him to the meetings his large loveable Airedale dog Sandy. Sandy would habitually choose a position on the rug directly in front of the middle of the blackboard. We who presented talks at the meetings were granted free use of the two ends of the blackboard, but the center piece was preempted by Sandy....

  38. The Perfect Parabola
    (pp. 133-134)

    Perhaps the most frequently told anecdote about Professor Coolidge occurred one day during his analytic geometry class, while discussing the conic sections. Coolidge had the habit of twirling his gold watch and chain, back and forth about his index finger as he lectured. At the start of a particularly vigorous swing, the chain broke, and the valuable watch looped across the classroom and landed, shattered on a stone window sill. Unperturbed Professor Coolidge immediately seized the lesson involved and uttered, “Gentlemen, you have just observed a perfect parabola.”...

  39. Three Coolidge Remarks
    (pp. 135-136)

    Professor Coolidge’s lectures were often enlivened with wit and humor. One day, in explaining the concept of a function passing to a limit, he stated, “The logarithmic function approaches infinity with the argument, but very reluctantly.

    Professor Coolidge had a penchant for probability theory. In an early 1909 talk, entitled “The Gambler’s Ruin,” which concerned the effect of finite stakes on the prospects of a gambler, he proved that the best strategy is to bet the entire stake available on the first turn of a fair coin. “It is true,” he concluded, “that a man who does this is a...

  40. Professor Coolidge During Examinations
    (pp. 137-138)

    Professor Coolidge used to pace around the classroom when his students were taking an examination. As he walked he would glance at the various students’ efforts. If his eye fell on something that displeased him he would point it out to the student and advise him to rework the problem....

  41. Professor Coolidge’s Test
    (pp. 139-140)

    I recall how, at a large mathematical gathering, Professor Coolidge rose, advanced to the front of the room, and there frightened the group by announcing that he was going to give them a little mathematics test. Now mathematics professors may like to give tests, but to take one is quite another matter. To calm his audience, Professor Coolidge said he merely wanted to verify that most mathematicians know very little elementary solid geometry.

    Professor Coolidge started by reviewing a few definitions, such as those of themediansand thealtitudesof triangles and tetrahedra. “Now,” he said, “though, as a...

  42. Borrowing Lecture Techniques from Admired Professors
    (pp. 141-142)

    I dare say that most of us who lecture in college tend to incorporate in our style some of the techniques we admired in our own former professors. I know I have done this. The two professors I most admired at Harvard were David Vernon Widder and Julian Lowell Coolidge, and I have amalgamated lecture techniques of these two professors into my own lecturing.

    Professor Widder was an absolute perfectionist. His lectures were meticulously prepared and delivered with great care. It was pure pleasure listening to him and following his inexorable progress through a topic. He was one of those...

  43. My Teaching Assistant Appointment
    (pp. 143-144)

    Although, when I was an undergraduate at the University of Virginia, I was occasionally invited to assist Professors Luck and Linfield, it wasn’t until my second year at Harvard that I became a genuine TA. The appointment was to assist Professor Ralph Beatley in one of his mathematics education courses. Professor Beatley was the authority at Harvard in the field of mathematics education. Not only did I grade the quizzes and daily homework assignments of his class, but I pinch-hitted for him when he was away on speaking engagements. This occurred quite frequently, for Professor Beatley was a much sought-after...

  44. A Night in the Widener Memorial Library
    (pp. 145-148)

    It’s pretty certain that no one would be able to guess how I was introduced to the fascinating study of linkage machines. Here is the story.

    We graduate mathematics students at Harvard were given keys to the extensive mathematics library housed in Room Q on the second floor of the university’s Widener Memorial Library. Room Q was a large square room lined with well-stocked bookcases, and furnished with a number of very comfortable reading chairs and some small writing desks.

    I spent a lot of time in Room Q, perusing many of the interesting books. I recall the delight I...

  45. The Slit in the Wall
    (pp. 149-152)

    On Harvard Square, back in the mid-1930s, there was a small thin eating-place frequented by four of us graduate mathematics students. We called itThe Slit in the Wall, and one day a surprising mathematical event took place there—in our favorite booth at the very rear of the place.

    We had each enjoyed a big bowl of delicious minestrone soup when I casually announced that I had discovered an “obvious” proof of a very teasing relationship in number theory that had been eluding us for some time. The word “obvious” caused a ripple of surprise among my friends, along...

  46. Nathan Altshiller Court
    (pp. 153-154)

    I was an early, and then a continued, admirer of Nathan Altshiller Court. He wrote one of the first textbooks on college geometry. It was an excellent book and it did much toward popularizing that subject among college mathematics offerings. I corresponded a great deal with Professor Court, especially during my long tenure as editor of one ofThe American Mathematical Monthly’s Problems Sections, for he was an assiduous contributor to that department. But it wasn’t until, upon a speaking tour, that I finally met him, at his university, the University of Oklahoma. We had a delightful visit together, and...

  47. An Editorial Comment
    (pp. 155-156)

    In the manuscript for the fifth edition of myAn Introduction to the History of Mathematics, I had written, in connection with the poorer Newtonian fluxional notation and the better Leibnizian differential notation, “The English mathematicians, though, clung long to the notation of their leader.” The copy editor deleted the word “long,” and wrote in the margin, “I thought Clung Long was a Chinese mathematician, not English....

  48. Intimations of the Future
    (pp. 157-160)

    I’ve often been asked what it was that caused me to become interested in mathematics—and just when and how did it happen. This will be told in a later story. Here I will tell of a couple of foreshadowings of that wonderful event.

    Some time in elementary school (I cannot now recall the precise grade) an excellent teacher introduced our class to the subject of geometrical areas. She first sketched on the board a rectangle 5 units long and 3 units wide. Then, drawing lines parallel to the sides of the rectangle through the unit divisions of those sides,...

  49. A Rival Field
    (pp. 161-162)

    Occasionally I am asked if there is some field I might have cared to go into instead of mathematics. Though there is a rival area that has strongly pulled on me, the conviction that I would have done so much more poorly in it than in mathematics makes the answer a definite “No.” Not only that, but I had already become dedicated to mathematics. That other area is music. My desires there would have been too high for me to attain, for I would want to composeoutstandingsymphonies—that, or nothing at all. So I content myself to listening...

  50. A Chinese Lesson
    (pp. 163-164)

    When I was teaching at Oregon State College (now Oregon State University), I designed and taught my first course in the history of mathematics. I entitled the courseGreat Moments in Mathematics. The title was motivated by a sequence of musically illustrated lectures, entitled Great Moments in Music, given some years ago on radio by the illustrious musicologist Walter Damrosch.

    My course consisted of 60 separate “great moments,” and each lecture was accompanied by an assortment of visual aids in the form of desk experiments, models, portraits, maps, overhead transparencies, and so forth. The course prospered very gratifyingly—to such...

  51. The Bookbag
    (pp. 165-166)

    The mention, in an earlier item, that I have lectured in every state of the union, except Alaska, and indeed many times in some states, brings to my mind one of the most prized gifts I have ever received.

    When I was a boy, my mother gave me an elegant bookbag, or brief case, made of sturdy artistically rippled black leather, equipped with three interior compartments, a strong handle, six little silver hemispheres on its bottom to protect it when it is set down, a little lock with a tiny key, and my initials H.W.E., in bright silver letters, beautifully...

  52. Running the Mile in Twenty-one Seconds
    (pp. 167-168)

    I have an identical twin brother, Don, and over the years we have had much fun confusing people—especially our teachers when we were in grade school together. Don’s field is biology, and for years he taught biology and general science in high schools and prep schools. He also possesses a quirk for harmless practical jokes.

    One fall day I visited Don when he was teaching at the Mohonk Prep School on Lake Mohonk in New York State. The school was operated in English style, with masters and forms instead of teachers and grades. When I arrived at his school...

  53. Winning the 1992 Pólya Award
    (pp. 169-170)

    Two planar pieces that can be placed so that they intercept chords of equal length on each member of some family of parallel lines, or two solid pieces that can be placed so that they intercept sections of equal area on each member of some family of parallel planes, are said to beCavalieri congruent. Some years ago I proved the following two theorems which, at first encounter scarcely seem to be true:

    Though one can exhibit two tetrahedra of equal volume that arenotCavalieri congruent,any two triangles of equal area are Cavalieri congruent.

    Though one can easily...

  54. A Love Story
    (pp. 171-176)

    When I was in the final grade of elementary school I fell madly in love. It was a clear case of love at first sight, and it evolved into an all-consuming passion of adoration that has continued during the entire rest of my life. I had fallen head over heels in love with the enticingly beautiful goddess Mathesis. Let me tell how it came about and then expand on some of its delightful joys.

    I, and my two brothers, were brought up in Paterson, New Jersey. On occasional Saturday mornings the three of us would go down Broadway to a...

  55. Eves’ Photo Album
    (pp. 177-180)
  56. A Condensed Biography of Howard Eves
    (pp. 181-182)

    In 1976 Professor Howard Eves retired from the University of Maine at Orono after a long and distinguished career as a teacher, geometer, writer, editor, and historian of mathematics. Since then he and his wife have divided their time between Lubec, Maine and Oviedo, Florida, with the choice appropriate to the season. From 1986 to 1991 he taught at the University of Central Florida in Orlando.

    Howard Whitley Eves was born in Paterson, New Jersey on January 10, 1911, one of identical twin brothers. Following graduation from high school in Paterson he attended the University of Virginia, graduating with a...

  57. An Abridged Bibliography of Howard Eves’ Works
    (pp. 183-184)