The Atomic Origins of My Scientific Curiosity
Leading to Mathematics

       As a child I was fascinated by electrical and mechanical things. My father (a molder) taught me to be a good mechanic and carpenter and how to solder wires; my parents provided me with an Erector Set (to build mechanical devices), a Lionel electric train, chemistry sets, microscope sets, radio kits,  etc. In 1940, when I was still 11 years old I read the amazing newspaper story below (related to the work of Otto Hahn and Lise Meitner in splitting the Uranium atom) and it opened up my total curiosity about the atom! My childhood imagination was absolutely fascinated by the fact that a small amount of this U-235 stuff could be so powerful, and I decided that I needed to learn physics and mathematics.

      My first radio set did not work. It was a mechanical imitation made from Erector Set parts; a blue, rectangular box with gearwheels for knobs. I built this when I was about 9 or 10 years old. Upon exhibiting it, I was told by the other kids: "But it doesn't really work!" This made me decide to figure out how to make a real radio set! I guess my first real vacuum tube radio set was made when I was 10 years old. I started reading every book in the public library on crystal and tube radio sets. I broke open old tubes to find out what the filament, cathode, grid and plate were. I unwound old electrolytic condensers (now called capacitors) to find out how they were made, and wound some of my own. I wound my own 5 volt, 20 ampere transformer to show I could do it. By age 17 I had built Geiger counter tubes (I learned glass blowing at a local Neon sign shop), Geissler tubes, an X-ray tube, a cathode ray oscilloscope, scale of 64 Eccles-Jordan flip-flop counting device, a radium spinthariscope, a giant 1,000,000-volt Tesla coil, Jacob's ladder, a mercury barometer, and I built my own working oil-drop apparatus a la R. A. Millikan to measure the electron charge, after reading his fascinating book "Electrons Plus and Minus".

       After reading about the invention of the transistor in 1947 by John Bardeen, William Shockley, and Walter Brattain, I made one myself around 1950 from a 1N34 germanium diode. This involved picking wax or whatever from a tiny hole in the side of the ceramic tube case, and using a microscope to insert a "whisker" wire very close to the existing whisker so as to form base, emitter and collector. It is this basic need to experiment, whether with wires or mechanical parts or numbers, that is the driving force of curiosity behind research, whether it be in mechanics, chemistry, biology, electronics or mathematics. My poor mother had to endure me stinking up our house as I tried to mix and boil sulfur and other noxious stuff in an attempt to make rubber. After spilling sufficient hot solder on my bed room floor, she sent me to our large unfinished attic which became my laboratory. There I had a 35 by 35 foot space, with a corner for radio work, a corner for chemistry sets, a corner for microscopes and biology, and a huge 15 foot bench my Dad made for my mechanical work and experimentation. This was my Walden, my hideaway, my place to read, study and think. In 1944-45 my giant Tesla coils, tossing 36 inch sparks and making terrible noise, and Jacob's ladders running yellow sparks up the walls, made the whole attic look like Dr. Frankenstein's lab. The house reeked of ozone. Neighbors could not get any radio reception when I had all my coils running. A nearby fireman would come out of his house with an ax and threaten to chop me down. And how many times did we electroplate old V nickels to look like five dollar gold pieces! How many small bombs did we make as we tried to make our own gun powder! And imagine all the nitrogen tri-Iodide we splashed around on key holes, door knobs and lamps to give everyone a tiny blast here and there. Maybe that was a little too much, but it was fun. My Dad was almost totally deaf, and when Mallory came out with the chunky, small mercury batteries, I saved every one of his old hearing aid batteries. You see, you drill two small holes in the copper top, and metallic mercury then gradually oozes to the top which you can suck out with a medicine dropper. I obtained a small Listerine bottle of Hg this way (cleaned by use of HCL) and used it to make a mercury barometer that works. This was raw, insatiable curiosity trying to find out how things work. It is like my student in a calculus class who tried to divide by zero because as she said, "When you want to figure something out, you will try anything!"

       It is interesting, however, that when I returned from army service in the early fifties, I found that my Mom had thrown out many of my tanks of acids and stuff for fear the house would blow up! She had also thrown out all my hundreds of old comic books, including first edition copies of circa 1938 Superman and the like (now worth a fortune), all my Astounding Science Fiction (ASF) magazines, and a hundred toy cap pistols! One excuse she gave was that science fiction "harms" the mind. There she was quite wrong. I have  even written some science fiction (unpublished) about time travel. John R. Pierce (1910-2002), who invented the traveling wave tube for microwave amplification, wrote for ASF under the pseudonym J. J. Coupling (derived from a quantum J to J coupling effect). In ASF Magazine he described his traveling wave tube as a true science fiction device invented before its useful time. But, of course,  he wrote his mathematical proof of the device in the Proceedings of the Institute of Radio Engineers (IRE).  Eminent combinatorialist John Riordan wrote fiction. Mathematician Eric Temple Bell wrote science fiction under the pseudonym John Taine. I have met many brilliant scientists who thrive on real science fiction, stuff about space and time travel, systems that depend on nitrogen instead of carbon, etc. I do not think my Mom ever realized what pure science is really about, any more than President George W. Bush understands stem cell research, or that global warming is a real threat to our civilization, or that creationism should not be taught in public schools!  What could be more like "science fiction" than "Flatland"?
. . . Nevertheless Mom and Dad supported me.

       My mother had started me reading by setting up a small library for me with hundreds of books on every imaginable subject, but heavy on languages (Latin, Greek, German) and science. By age 10 I had learned the Hebrew alphabet, and I began to study written Chinese. Mom taught me etymology at home using an old book that traced the origins of English to its Latin, Greek and Anglo Saxon roots in great detail. I was exposed to the curious old book by Albert C. Crehore on the theory of the atom, where I first learned about the infinite series I later learned is called the Riemann Zeta function. The book introduced me to the Bohr theory of the atom. I pondered the possibility of the existence of Element 118. This was in the years before Neptunium and Plutonium. In college in 1946 I started learning some Chinese and Arabic from textbooks written in German. 

       In high school days there were three of us (Paul Louis Goodfriend, Bernard Brown and myself) who were the genius science kids. Around 1944-45 a Los Alamos physicist showed us the (then-classified) Los Alamos Report LA-24 which outlined atomic theory and gave the differential and integral equations of neutron flow in an atomic pile. We studied Pollard and Davidson's book "Applied Nuclear Physics" and Strong's "Experimental Physics". We absorbed the "Smyth Report". We studied Uranium chemistry, and "We Three" set about to extract one gram of Uranium from pitchblende ore or Uranium glass (used in neon sign manufacture). There we were, three high school kids mixing hydrofluoric acid with uranium glass in lead crucibles, and ultimately going through a flow chart to obtain a small quantity of Uranium Tetrafluoride and ending up with a tiny amount of Uranium Oxide (U2 O8), which is what Klaproth obtained when he discovered Uranium in 1789. In the course of our crude experiments we probably breathed in a lot of the volatile Uranium Hexafluoride vapors from our crucibles! Thank our lucky stars we did not suffer! By the way sodium diuranate (Na2 U2 O7.6H2O), also called yellow cake, and uranyl nitrate (UO2(NO3)2) don't taste bad at all! Brown, Goodfriend and I studied the theory but were not able to obtain some critical items to build a linear accelerator or a cyclotron.

       My late friend Jerome Hines (Metropolitan Opera Star who sang 30 roles at the Met) started his studies in mathematics, physics and chemistry. But his voice was better, and he soon became a basso profundo singing sensation.  Dad took me to hear him sing many years ago. I tried singing but gave it up for science, just the reverse of what Jerome Hines did. My Dad had a really good voice but no training as his brother William Benjamin Gould had had. Uncle Bill studied voice at the Toronto Conservatory after the First World War and sang opera in Europe and especially in Moscow. He had a White Russian girl friend and wrote a novel "When White was Red" but this was never published, being critical of the political system under Lenin.

       Curiously, Hines and I  had a mathematics professor in common! William Marvin Whyburn was Jerry's mathematics mentor at UCLA and many years later I knew William at UNC in 1957. This Whyburn was the brother of the famous Gordon Thomas Whyburn, topologist and student of R. L. Moore. Jerry published about 12 papers in the Mathematics Magazine; one was on the Stirling numbers. Hines visited West Virginia University several times, singing and reviewing performances of master's voice students. One time when Jerry visited here at Morgantown, the Maestro reviewed voice students in the morning, then we had lunch, and in the afternoon the Maestro gave a talk in our Department's Colloquium about "Operator Mathematics" (see the program poster under "Gould's Photo Album" here on this web site). The first time he sang here, I had him autograph reprints of his mathematics papers while everyone else wanted him to sign the concert program. Jerry and I sat up till 3:00 AM at the Hampton Inn here in town reminiscing about mathematics, chemistry, physics and music. He told me that he too had dabbled with uranium and that he once had his own small bottle of uranium nitrate with which to play. This man, this great basso profundo, was the epitome of the Renaissance Man Ne Plus Ultra. He once gave a short course at the University of Illinois on "Calculus for Poets". Such was his enthusiasm for the beauty of mathematics. Baritone Joseph Shore has a web page <> giving a marvelous memorial to Jerome Hines.

       Circa 1945, using the greatest integer function (bracket function), factorials, and other stuff, I was able to devise a formula for the determination of the binding energy of every known isotope of every then-known element! I sent it to a famous physicist but he never answered my poor letter, and I did not know until years later how accurate my formula was. It remains unpublished. Here is advice to teachers and professors: Always answer inquiring letters from budding scientists. I understand that Einstein (not the professor to whom  I wrote) always answered such letters from youngsters. I know (from personal observation when he visited three times here at West Virginia University) that the late number genius Paul Erdös was always eager to interact with bright youngsters, giving maximum support for their curiosity. Genuine curiosity must be stimulated and encouraged throughout one's lifetime. Without curiosity and enthusiasm there can be no advance in science and our understanding of the Universe. Blind and unquestioned obedience to religious myth stifles the search for knowledge about the universe. This is not say that religion is 'wrong'; indeed the Bible (whichever of hundreds of translations you choose) is a document about morality and ethical behavior, and has no meaningful information about science or how the physical universe came to be and does not refute the truth of evolution. There is no conflict between religion and science.

       In high school I read in Popular Science Magazine (July 1944) about the maverick Viennese physicist Felix Ehrenhaft (1879-1952) and his claim to have discovered a magnetic monopole. I built models of his "magnetic motor" and pondered what his claims about a "magnetic current" meant.  Of course at that time I had not studied enough physics to understand this. But doing these experiments was bold, raw curiosity in action! Ehrenhaft was very controversial and disagreed viciously with Einstein.

       Now, already in 1945 I had published a circuit diagram for a reflex regenerative radio receiving set in a radio magazine (Radio Craft Magazine) and again in 1948 a circuit design for a super-regenerative FM radio receiver.

       My first mathematical discovery (around 1942) was the formula n(n-1)/2 for counting the number of pairs of pins you must choose to check for continuity when testing a radio tube with n pins on the base. Of course this also led to the explicit determination of the actual permutations and combinations of  k  items chosen from a set of  n  items.

       Around 1944-45 I developed a formula for finding the sum of the k-th powers of the first n natural numbers. Only later did I learn that I had "rediscovered" the Bernoulli numbers. This was my first experience with recurrence formulas or recursive algorithms to prove an interesting result by mathematical induction.

       I decided that I would make a project in my life to study everything knowable about sums and products of numbers. My algebra teacher Miss Elizabeth Culpepper, at Woodrow Wilson High School in Portsmouth, Va., had let me study factorials and binomial coefficients while the other kids were studying the quadratic equation, and I became utterly fascinated by the integer coefficients when you multiply out  x(x - 1)(x - 2)(x - 3) . . . (x - n + 1), and it was in this way that I became intrigued with what I later learned are the Stirling Numbers of the First Kind. I studied them in my master's thesis (1956) and later published a simple formula for them in the Proceedings of the American Mathematical Society (1960). After joining the faculty of mathematics at West Virginia University in 1958 I was able to obtain NSF grants starting in 1960 to support my research in combinatorics. Later I found out that these were the first NSF mathematics research grants in the State of West Virginia. I made a trip back to my home town to visit my high school algebra teacher to thank her for encouraging my fledgling interest in permutations and combinations. It was because of her encouragement that I stayed with mathematics and I credit her for edging me on toward success in combinatorics.

       Two other high school episodes will show other reasons for my plunge into science, and especially mathematics. I remember a time circa 1942-44 when my English teacher asked our class to write a theme (i.e. an essay) on something we were really interested in. Ergo! I wrote my essay "Introduction to the Mathematical Theory of Four-Dimensional Geometry." I had been reading "Flatland" by Edwin A. Abbot. I had also absorbed Henry P. Manning's "The Fourth Dimension Simply Explained" (1910/1912) and his  "Geometry of Four Dimensions" (1914). These were among the various books that my mother had provided me. My teacher gave me a grade of C minus, and I asked her about the grade: "I can understand the C because I am just an average kid, but why the minus?" Teacher's answer: "Poor choice of topic!" Then I understood what she really meant: We could write on any topic we wanted . . . provided she could understand it! Then I heard a mathematics teacher tell another teacher: "Don't let that Gould kid in your class; he asks too many questions!" . . . ah, but that is the only way I learn, by asking questions. At ten years of age being told that God created the Earth, my response was: "Then who or what created God?" I asked: "How many years existed before time was created?" I was told quite summarily that I couldn't ask such questions. Ah, but you see, I did and do ask such questions.

       I remember a Greyhound Bus trip to visit Professor Florence Mears at George Washington University because I saw in a college catalogue that she taught a course on infinite series. She and others (e.g. Saunders Mac Lane at University of Chicago) soon made it clear to me that I would have to study a lot more than a simple course on infinite series in order to achieve my dream of summing all the series!

       Most importantly, I wrote to Leonard Carlitz at Duke around 1953. He liked some of the formulas I sent him but some of my stuff had been anticipated long ago by Ramanujan. By analogy, Carlitz became my G. H. Hardy, and he taught me so much that I shall forever be indebted to his infinite patience with my finite brain!

       It should be remarked that my mother left me a letter which I found after her death on 27 July 1964, in which she explained that she had prayed for a child for 13 years. She was nearly 39 when I was born, and I was the only child. It is a difficult task to be an only child and live up to your parents' expectations. The Bible commands us to honor our father and mother. All that I am I owe to my parents.

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New Chemical Discovered

New York, May 4, 1940 - (AP) - Laboratory isolation of a new chemical substance, one pound of which is said to be capable of yielding the power output of 5,000,000 pounds of coal or 3,000,000 gallons of gasoline, was described today by the New York Times.
    The Times said that the discovery, announced in the current issue of the Physical Review, scientific journal, had been hailed by leading scientists as holding the promise of revolutionizing all present methods of power production, and ushering in the era of atomic power.
    The substance was identified as "U-235", an isotope or chemical twin of ordinary uranium, which when simply immersed in cool water releases its energy in a form unseeable by man ---- steam.
    A chunk of 5 to 10 pounds of the substance, plentifully available in many parts of the earth, would drive a battleship or sea-going submarine around the oceans of the world for an indefinite period without refueling, it was said.
    The Times said that the Nazi government had heard of American research in this field and had ordered its greatest scientists to concentrate on the problem of improving the method of extracting the U-235, one pound of which was said to have the explosive force of 15,000 tons of TNT.
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       from: The Portsmouth Star newspaper,
                 Portsmouth, Virginia, Sunday, May 5, 1940, Local Section, page 8.

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Addendum about Number Theory

       When I was in Army basic training, I carried around a list of all the prime numbers less than 10,000. I had factored the serial numbers of all the other guys in the company, but could not factor my own number. Whenever we were out on a long march and were given a "five-minute smoke break", I would dig into my back pack and get out my list of primes and continue to divide my serial number by the next prime, always getting a remainder, so I finally decided that I, the only mathematician in the company, was a PRIME! Years later, first using a huge mechanical adding machine, then an early desk-top computer, and finally a big IBM-360 digital computer, I was able in microseconds to convince myself that I was indeed a PRIME. It was not until around 1962 that all of the first 100 Fibonacci numbers were finally factored or shown to be prime. There is this big unsolved problem to show whether there are infinitely many prime Fibonacci numbers. So far as I know this is still undecided.

       In the first issue of the Fibonacci Quarterly, Volume 1, February 1963, page 46, Leo Moser and Leonard Carlitz posed the second advanced problem H-2 "Resolve the conjecture: There are no Fibonacci numbers which are integral squares except 0, 1, and 144." The same problem was posed in the American Mathematical Monthly, Vol. 70, No. 2, February 1963, page 46, Problem 5080, by A. R. Rollett, Crediton, England, in the form: "In the Fibonacci series (F_1 = 1, F_2 = 1, F_n+1 = F_n + F_n-1) the first second and twelfth terms are squares. Are there any others?" Between the two journals, I believe about 7 proofs were received (not all published) showing that 144 is indeed the last Fibonacci square. In the meantime Marvin Wunderlich had run computer tests showing that no other squares existed in the first million Fibonacci numbers. In time it was been shown that 8 is the last cube in the sequence, and it appears that someone has now shown that no other perfect powers appear in the Fibonacci sequence. These are difficult topics.

       In a lighter vein, I posed the very first problem (H-1) in the Fibonacci Quarterly, Vol. 1, Feb. 1963, p. 46: "Find a formula for the n-th non-Fibonacci number, that is, for the sequence 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23,  . . . " I gave hints to see a paper by Leo Moser and Joachim Lambek, and later published a paper on this in the Fibonacci Quarterly, Vol. 3(1965), pp. 177-183. with the title "Non-Fibonacci Numbers." This was all about complementary sequences whose intersection is empty but union is all the natural numbers. I used to joke Vern Hoggatt that I would have to start a Non-Fibonacci Association and publish a Non-Fibonacci Quarterly. It appears that the study of complementary sequences began in earnest with Samuel Beatty in Canada, and was long a popular Canadian topic.

- - - -  Henry W. Gould, 26 August 2008