Gabriel Kron

Gabriel Kron Photo and © Union College Press, Schenectady, 1973

Gabriel Kron was born in 1901 in Nagybanya (Baia Mare)in the Carpathian Mountains in Hungary. He and his brother reached New York in 1921 as immigrants. Both brothers entered Michigan Universitiy the same year after learning English the fast way. Gabriel Kron later took a job with the Lincoln Electric Company in Cleveland and presented his first paper in 1930. 1931 with Warner Brothers. In 1934 he joined General Electric and worked there in various departments, all concerened with applied engineering. Doctor of Engineering - 1936 : note of Univ. Michigan. His books and 100 papers were written on weekends and at night. Gabriel Kron retired in 1966, died from a fatal illness in 1968. - - - Some lively and interesting biographies beginning here. - - - Gabriel Kron's List of Publications here.

Gabriel Kron 1901 -1968 The following short note about Gabriel Kron was found in the internet 8/2003, precisely here. .... Casazza: Gabriel Kron, a very fascinating man. What I could do with the history of Gabriel Kron. He was thrown out of the University of Michigan. I'll tell you a little bit about him that's not in my book; I did something else. He was thrown out of the University of Michigan because he was always fighting with the instructors, at something like sixteen. He decided to work his way around the world, and came to Hollywood. He was very brilliant. He had so many problems because his professors were a couple of light years behind him. He got back to Hollywood, signed a contract for $10,000 or so to work on his new experimental movie camera, and the company that gave him the contract paid him the money up front and went bankrupt. So he had a year or two with no work to do. He came to New York City. In the public library he started to read books on mathematics and became the inventor of something called tensor analysis. It became quite important but then he worked for GE. He was unusual and was not easy to work with because he was ahead of his time. You have to mention him in the history of electrical engineering because he was a character.... Book listing (no ad) taken from alibris.com 8/2003. Yet another reference to G. Kron, 8/2003, - from here ..... Andrei Petrov described Kuznetsov's work on the method of tensor analysis for the handling of physical systems of extreme complexity, based on earlier work by the American engineer Gabriel Kron, whom Kuznetsov held in high esteem. Petrov also recounted the origin of the discovery of the significance of what Kuznetsov called the "Principle of Conservation of Power," for the understanding of living systems as well as physical economies, whose evolution proceeds in the opposite direction as that implied by the so-called Second Law of Thermodynamics. ... Also: Gabriel Kron. Tensors for Circuits. Dover Publication, Inc., second Edition, 1959. Some other links: http://www-ec.njit.edu/civil/fabric/automated/CH9.pdf

PHILIP LANGDON ALGER - (1894 - 1979) - was graduated from St. John’s College of Annapolis, Md., in 1912 and from the Massachusetts Institute of Technology in 1915 with a B.S. degree in electrical engineering. He earned the M.S. degree from Union College in 1920. St. John’s awarded him an honorary M.A. in 1915, and the University of Colorado an honorary Doctor of Engineering degree in 1969. He worked for General Electric Co. as a designing, staff, and consulting engineer until his retirement in 1959. From 1959 to 1969 he was Consulting Professor of Electrical Engineering at Rensselaer Polytechnic Institute. The author of more than 100 technical papers and three books, Malhematics for Science and Engineering, Induction Machines, and The Human Side of Engineering, he edited the book, The Life and Times of Gabriel Kron. In 1959 he received the Lamme Medal of the A.I.E.E.



GABRIEL KRON, { this website: from here on in the original 'Gabe' } the youngest child in a family of eight children, was born on July 23, 1901, in Nagybanya, later renamed Baja Mare, a town of about 10,000 people in a remote region of the Carpathian Mountains of Hungary. His father, the owner of a dry goods store, was a devout follower of the Jewish faith. In writing of his family life, Kron said:

"Ever since I can remember, every night at bedtime father used to sit at a table in the center of the room, in front of him a large Hebrew book, covering half the table. He sat there, slowly swaying backward and forward, caressing with his hand his long whiskers, or pointing with his index finger to his temple. For hours and hours he kept bowing to and fro, reading and studying the Law in a peculiar melodious singsong brought down through the ages.

"Every day of the year, winter and summer, father got up at five o'clock in the morning and until six o'clock he continued to study while I lay in bed half awake and half asleep, following his train of thought by simply listening to the rising and falling cadence of his tone, without understanding a single word of it. And my thoughts would slowly wander off into the land of fancy. I saw myself climbing across the Himalayas into the forbidden land of Tibet; I built skyscrapers in New York City or erected power plants at the foot of Victoria Falls in Africa. As my fancy wandered from charted to uncharted territories, I would attempt to transmit electric power without wires or build rocket planes that could fly to the moon or distant planets. I was a physician who discovered a remedy for some incurable disease or a physicist who could transmute the elements." (1)

A boyhood friend described Gabe in elementary school as plump, always lively, always joyous, always having something to tell. He added, "Gabe inherited the vivacity of his character and the strength of his resolutions from his mother. When still a boy, he formed a realist's Weltanschauung and decided the way he should follow in life. At that time people who knew him well understood quite clearly that for him the environment of his home town was too confined and that a day would come when he would act to broaden his horizons."

His passion for knowledge became evident in early school days. His teacher was accustomed to ask for drawings, and Gabe made the greatest number of these. His drawings were chiefly maps of different countries, taken from all the continents. He was quite sure he would soon pay a visit to each of them. In the Gymnasium { this website: a German & Austrian word, (no connection whatsoever to the engl. word 'gymnasium' ) about equivalent to: high school / first college classes } he studied intensively physics and mathematics; he also devoted himself to astronomy, stenography, and the English and German languages. At sixteen, he remarked that there were no more books in the library for him to read. On the lighter side, he was an editor of the school paper, and he took a leading part in staging a most successful cabaret show.

Kron realized even in the lower grades that the Hungarian language would be useless for one who wanted to specialize in the physical sciences. He said in later years, "Being subjected during my life time to several inoculations with a new language, I gradually developed a technique of my own; that proved to be quite practical for acquiring a reading knowledge of a new language in a place which offered no contact with people speaking it. As young children we had been instructed in Hebrew by being handed the Bible and asked to translate it without the aid of dictionary or grammar, merely being told by the teacher the meaning of each word. Each of the pupils recited and translated a verse in succession, each one thereby getting a chance to try his knowledge and luck on the tenth or fifteenth verse. Rarely, of course, did we follow attentively the translational efforts of the others, and we gained only the haziest notion of any connected story.

"Ever since I was six years old I had gone through the Bible - or a portion of it - once a year, translating it in the above manner. When I reached age fourteen I gave up the whole struggle as hopeless and an absolute waste of time (also because at fourteen I could resist successfully my father's will). In the Gymnasium we were taught Latin for eight years, German for six, and Greek for four years, each with the aid of the most elaborate grammar the human mind could devise. Under the guise of classical education the professors crammed our heads with an infinite variety of useless rules, exceptions to the rules, and exceptions to the exceptions, so that languages were the most lugubrious subjects taught....

"Realizing that the manner in which languages were taught was one of the hundred possible ways that should not be employed, I started out to use just the opposite tactics. Instead of memorizing grammar, I memorized the German dictionary. Our school dictionary had about twenty thousand words in it, a hundred words on a page; so, I tore out a page a day and committed it to memory. Then the page was thrown away. When there was no dictionary left, I considered myself an expert in the language. I could sit down, open up a French or German book anywhere, and proceed. Slowly and painfully at first, but nevertheless I could make headway in understanding the text.....

"It rather disappointed me to realize that familiarity with German and French would not be of much practical use in my attempts to relieve Europe of my presence. I sensed - in spite of the arguments of those who had travelled far and wide - that once Europe receded behind my back the only language that would enable me to move about freely must be English. Unfortunately (or fortunately) no English-Hungarian dictionary could be procured in all Transylvania, and the only one I could get hold of was a big Muret-Sanders type of English-German dictionary, containing over a hundred thousand words. Of course it was out of the question to repeat the stunt of tearing out the pages and memorizing them one after another, as they contained so many archaic and technical words. What I did was to borrow an English book, which happened to be H.G. Wells' The Food of the Gods, and beginning with the first page I wrote out a hundred words each day and committed them to memory. After eight or ten pages the text began to assume more human form, and past the first chapter the content even became enjoyable." (2)

Before World War I private tutoring was an old custom in Hungary. Every well-to-do parent aspired to have his sons finish the Gymnasium, for no one was considered a gentleman unless he graduated from the Gymnasium, and no amount of money could compensate for the lack of this credential. A sizeable dowry, however, was sufficient to create a lady. It was not considered proper for girls to attend the Gymnasium, so the few girls who sought an education relied on tutors. This gave the poor boys their opportunity. Since classes ended every day by 12 or 1 o'clock, many young men spent the afternoons tutoring. During Kron's upperclass years he used to spend from 2 to 9 or 10 o'clock each day in this way, giving on the average an hour and a half to each pupil.

In the summer of 1918, he was invited to live with a family in Felsöbánya, seven miles away from home, to tutor two girls. He was so successful that he continued to live there during the eighth, and final, year of the Gymnasium, even though he had to walk fourteen miles each day to go to school and return.

In the fall of 1918, however, the Rumanians took over that part of Hungary, peacefully and without ceremony, and they turned the Gymnasium into a cavalry stable. School stopped, and the principal advised the students to study at home and prepare for a private examination next June. So Kron stayed at Felsöbánya, rearranging his schedule to spend the mornings in teaching the girls while the afternoons were devoted to his own intellectual needs. From 1 to 2 o'clock he practiced Hungarian shorthand, from 2 to 3 he solved algebra problems, from 3 to 4 he studied French, from 4 to 5 physics, from 5 to 6 translated German, and so on up to 10 in the evening. He was then seventeen years old.

With the closing of the Gymnasium, and the rule that the future classes must be conducted in the Rumanian language, which he did not know, Kron realized that his future lay in foreign lands, and the search for that land occupied all his thoughts. His oldest brother came home from the war, bringing fresh energy and ideas to the conduct of the family store, so Kron hoped for a time that the family might finance his going to England to study. But thieves broke in and robbed the store, making that plan impossible.

The Rumanian Minister of Education allowed the Gymnasium to reopen for one month to conduct final examinations for Kron’s class, so Kron gained the coveted diploma in June 1919. He then formed a new plan. Another older brother, Joseph, who was wounded in the war, came home with a little money and a strong desire to gain an education. He had dropped out of school after only four years, saying: "I am not interested in knowing what the stars are made of." Gabriel appealed to his brother to join him in going to America to study engineering, and he promised to tutor the older man so that he could earn his Gymnasium diploma in a single year. Joseph agreed and, with Gabriel's help, obtained a full set of the books studied in the eight years of the Gymnasium, plus a permit allowing him to take examinations whenever he was ready. Starting in late October 1919, Joe began intensive studies. Gabe selected only 10 per cent of the pages from the books - sometimes less - and Joe learned only those. In January Joe passed examinations for the first four grades, in April he passed the fifth and sixth, in June the seventh, and eighth, and in August he passed the maturity examination with a better-than-average B grade, thus earning his diploma in record fashion.

The whole family now gave its support to the two young men. Gabriel and Joseph Kron sailed from Antwerp on the SS Mongolia, and in January 1921 they reached New York.

They were welcomed by the Hebrew Immigrants' Aid Society and began at once to search for jobs. Gabriel became a dishwasher in a Hungarian restaurant on 116th Street, earning $5 a week and meals, working ten hours a day, seven days a week, with one afternoon off. Joseph found a job in a fur shop. Each evening, Joseph was expected to memorize fifty new words from the dictionary, and both began to attend free lectures to improve their English. Gabriel went to the public libraries and looked up universities (a college or an Institute had no standing in European eyes), seeking an engineering school that would cost no more than $150 for tuition, located far enough inland to be away from the flood of immigrants. The choice fell on the University of Michigan. He sent their credentials to Ann Arbor, and the brothers were accepted to enter in September. It was essential to earn more money now, so Gabriel began an intensive search for a better job. After trials at necktie peddling, botling vinegar, sign painting, and steam-pressing knitted ties, he found a job as a busboy in an Automat at $15 a week plus meals. With this princely income, and by sharing an apartment with some Hungarian students, they bought new suits and saved $150 each, enough to pay their carfare and the $125 tuition.

In Ann Arbor, they had first to find jobs that would pay for food and rent. For two months they lived on bread, milk, and grapes, but they found a room off campus where the rent was only $3 a week, and they sold their student football tickets for some $40. Gabriel worked for his meals, but no pay, as a dishwasher in the Michigan Union; his brother found a job in a downtown fur shop. In the summer Gabriel became a ditchdigger at $24 a week. Making their way by these varied methods, they also organized their classwork to suit their needs. While Gabriel learned the daily assignments, he underlined the parts that his brother should study; sometimes, when Joseph worked overtime or didn't know the answer, Gabriel answered for him - until the professors learned to distinguish between the "Big Kron" (3) and the "Little Kron." The latter wrote: "As always happens when a free spirit is obliged to undergo a prescribed routine, I wanted to study everything except what the curriculum called for. How to find time to study what one wishes, and not what the teacher thinks best for one's own good, must be a perennial problem to many an anxious pupil. Finally, I hit on the idea of arranging my schedules so that by Friday noon the classes would be all over. Three full days then each week from Friday noon to Monday morning I was free to pursue my own private schedule of study without the interruptions of regular classwork. The rest of the week I considered as a sacrifice on the altar of mechanized education." (4)

At the beginning of his junior year, it became clear that Gabriel Kron would complete the requirements for graduation by the next June, so there would be no senior year for him. He began to think what to do next, and his fancy turned to his early dreams of world travel. In the course of his dreaming he wrote a small essay, "Out- line of a New Cosmology," describing the universe as an engineer might have built it, disregarding small obstacles like the law of gravitation and relativity. He showed it to a lofty professor and received the advice: "Young man, if you want to be a scientist, never try to introduce anything that has not been approved first by a committee of a scientific society."

The head of his department invited Kron to return to Michigan as a teacher, but he declined. And the General Electric interviewer did not offer him a job, as he seemed not to fit any of the available openings - he was too unique.

The more he considered it, the more he liked his old plan of walking around the world - it would be a new type of graduate study. The customary post-doctoral study in a German university could as well be replaced by overnight discussions with the monks of Tibet. He announced to his friends that he would put on a knapsack and start on foot around the world. Who would go with him? With no takers, he put an advertisement in the Michigan Daily, asking any one interested to telephone him. No one called. With graduation past, he went to work for four weeks digging sewers to earn the money needed for his world trip. After buying necessities, he had $28 left, and on a Monday afternoon at 3 o'clock he set out on the road to California. The full story has been told elsewhere, (5) so it will suffice to sketch just an outline of his experiences.

When he reached Los Angeles he went to sleep under a eucalyptus tree with just seven cents in his pockets. He found employment with the U.S. Electrical Manufacturing Company, however, and began his engineering career by designing induction motors. Soon he transferred to the Robbins and Myers Company in Springfield, Ohio; there he worked for W. J. Branson, who proved a most sympathetic and capable guide. (In 1938 Kron dedicated his book, The Application of Tensors to the Analysis of Rotating Electrical Machinery, to Branson.) But when his citizenship came through in 1926, he started again for California. There he took passage on a tanker for Tahiti with $300 and Forsythe's Differential Equations in his knapsack.

He lived as a guest in the family of a native in Tahiti for some weeks, then sailed to the Fiji Islands. He walked through the back country, shared native hospitality, swam rivers, and admired the tropical forest. He wrote: "Paradise could not possess a more luscious and exorbitant panorama of luxurious vegetation. Had I been a painter, I would never have left the island [of Viti-Levu]." (6)

In all his travels, his custom was to spend the usual siesta hours under a tree, studying mathematics. Having finished his study of differential equations in Fiji, he buried the Forsythe book in an empty oil can under a large tree, dedicating it to the memory of the early missionaries who had been eaten by the natives.

Then he took ship to Sydney, where his money ran out, so he found a job in a one-room plant engaged in making watt-hour meters. Kron was employed to develop a thermostat: After working long enough to clear £35, be bought Weatherburn's book on Vector Analysis and took to the road again, hitchiking to Townsville in Queensland. On the way he spent many nights in the company of the "sundowners," the Australian tramps who lived without working, with the aid of free food tickets handed out in police stations to anyone who needed them.

He wrote: "In those long weary walks through Queensland and later through Asia the outlines of a many-dimensional vector analysis began to take shape in my mind. Under the stimulation of my everyday preoccupations with imaginary maps of unknown territories, an analogous mental picture of engineering structures - such as an electric machine, or a bridge, or an airplane - engraved itself in my mind. They appeared (for purposes of analysis) as a collection of numerous multi-dimensional spaces connected together into one unit in a manner very much as the numerous countries and islands and continents are interlinked by a web of roads and customs and laws.

"If communication between the various members disappears, nothing physical is lost, only that intangible something that transforms the forty-eight independent states into a single U.S.A., or the many thousand independent parts into a single airplane. Years later I discovered that mathematicians had already laid a firm foundation under the name Tensor Analysis for just such a type of calculus as I attempted to develop." (7)

Kron sailed from Townsville to Borneo, and thence to Manila, Hongkong, and Saigon. Here, he started out on foot to Angkor Vat, and walked on to Aranha, where he took a train for Bangkok, then joined a caravan that followed the ancient trade route to Cockrake in Burma. He walked to Rangoon, took a boat to Calcutta, walked on to Agra, where he admired the Taj Mahal. He crossed the Indian desert to Karachi by train, took a boat across the Persian Gulf and went on by train to Baghdad, stopping to see the ruins of Ur on the way. He spent $5 for a truck ride across the Arabian desert to Damascus, then set out on foot again to Gaza. He hastened on to Cairo by train, saw the Pyramids, sailed from Alexandria to Constantinople and went by train to Bucharest, arriving at midnight at the home of a friend, with just small change left in his pockets. After an all-night talk, the friend financed his ticket to Baja Mare, where his parents welcomed him. The neighbors wanted to know why he had come home from America with his toes sticking out of his shoes, while every one else came home rich. No one asked about his travels.

Kron spent several months at home, studying and also courting his future wife, Ann, before returning to the United States. He took a job with the Lincoln Electric Company in Cleveland in the fall of 1928, and in 1930 he presented the first of his more than a hundred technical papers.

That first paper, entitled "Generalized Theory of Electrical Machinery," was the beginning of his series of papers presenting more and more comprehensive analysis of machines and systems. He thought that all types of electric machines must be special cases of a generalized machine, and that understanding the general machine would lead to the invention of new types. After transferring to the Westinghouse Company in Springfield, he presented his second A.I.E.E. paper in 1931. It explained the effects on induction motor torque due to field harmonics (superposed multipolar fields) in the air gap. Since my 1920 Union College thesis dealt with the same topics, I was much interested and took part in the discussion of Kron's paper. This was the beginning of a friendship that kept us in close touch for more than thirty years.

In 1931 he accepted an invitation to join Warner Brothers to design phonograph motors, with a three-year contract at $10,000 a year. The Depression forced the closing of his department, however, and for some two years he had a fine salary with no duties to perform. He resolved to devote this free time to study that would lead to a much better job after the Depression and, meanwhile, to economize by living at home in Baja Mare. So he and Ann returned to Rumania for a year. He read widely in the field of mathematics, becoming familiar with tensor analysis and Non-Riemannian geometry for the first time. Seeing an analogy between these abstruse concepts and the complex interrelations of electric, magnetic, and mechanical forces in machines, he wrote his classic paper on the "Non-Riemannian Dynamics of Rotating Electrical Machinery," which won for him the Montefiore Prize from the University of Liege in 1935. He sent me a copy of this paper in 1933, when he returned to the U.S., and I was so impressed by it that I sent it to my friend, Professor Philip Franklin, of the Mathematics Department at M.I.T., who was the Editor of the M.I.T. Journal of Mathematics and Physics. ‘He published the paper in full in the May 1934 issue of the M.I.T. Journal.

The paper instantly produced wide-spread discussion and controversy. Working alone, applying mathematical concepts in ways never done before, Kron gave new meanings to terms and disregarded established rules, so that many mathematicians derided his work: it was just for show, it was needlessly complex, or it was of no practical use. When he first put his ideas forward, there were no large computers, and engineers were little concerned with systems, so it took some years for their value to be appreciated. But Kron's theories, derived largely by intuition, have been proved sound and are increasingly useful. Instead of calculating the effects of changes A, B, and C separately, and then finding that each change required the others to be recalculated, Kron's methods take all the interrelations into account at once, thus opening the way to correct analysis of the most complex systems.

To throw more light on the new ideas, I arranged an A.I.E.E. conference in January 1934, in New York, at which Kron presented his views to interested engineers and some pure mathematicians. He did so well there that I urged General Electric to employ him. After checking with his Michigan professor, A.D. Moore, and others, Vice President Roy C. Muir invited Gabe to join the staff of the Advanced Engineering Program, under A.R. Stevenson, Jr., in May 1934. He did so, and began at once to extend and deepen his studies of electric machines, power systems, and computers. The results poured forth in five books and a hundred technical papers during the next thirty years.

The Advanced Engineering Program staff at that time included the mathematicians E.0. Keller and H. Poritsky; a very distin-guished engineer, C.A. Nickle; and a young engineer who gained distinction later on, Loyal V. Bewley. Bewley was a graduate of the first class of the Advanced Engineering Program, who took a deep interest in the ideas put forward by Kron. In later years Bewley criticized Kron's writings, saying that he used too many words, did not make his points clear, and made simple things appear complicated. However, this may be, Kron was too absorbed in pursuing his own widening train of thought to be interested in backtracking or in refining his earlier work. He was a pioneer, not an educator: He used to imply that hard work is required for mastery of any subject, and it does no good to make the way to understanding too easy. He was doing the really hard work of breaking trail, and those who came after him could follow well enough if they were truly capable. He told me at one time that he felt as if he would burst - so many books were welling up inside him, calling for him to move on to unexplored fields.

One difficulty was that Kron always wanted to have his theories apply widely - he always wanted to generalize. Thus his methods were more complex than required for any particular problem. They could be used for a wide variety of problems, but the engineer who was concerned only with a particular machine preferred to use the simplest methods and took little interest in Kron's elegant theories.

Stevenson left Kron to pursue his own ideas but kept him in contact with the students in the program. Kron was a great talker and could hold his hearers spellbound, whether or not they understood him. In this way, he inspired many students to think more broadly. One student in particular, Charles Concordia, began to apply Kron's ideas to practical purposes. After finishing the course, Concordia went into the Apparatus Sales Department, where he worked on problems of power supply and transmission. In considering extensive power networks, with large numbers of supply stations and load centers, he felt the need for more general methods. To help in power system analysis, Kron also transferred there in 1939. As computers became available, Concordia and Kron did a great deal to establish the modern methods of system analysis.

Kron spent nights and weekends pursuing his own thoughts. The results appeared in an extensive series of articles published in the G. E. Review between 1936 and 1942. He also kept in touch with graduates of the Advanced Engineering Program who had gone to work in various departments of the Company, and helped them in many ways, particularly by developing correct equivalent circuits for all kinds of machines and systems. These were published in his third book, Equivalent Circuits of Electric Machinery, in 1951. His first two books, Tensor Analysis of Networks and A Short Course in Tensor Analysis for Electric Engineers, appeared in 1939 and 1940, respectively.

In 1942 Kron transferred to the Large Steam Turbine Engineering Department, to work on problems of stress analysis in complex steel structures. Then he moved again, to work on electronics with Simon Ramo. And, in 1945, he moved to the Research Laboratory, where he worked on the temperature distribution and control of piles for atomic reactors, and other abstruse problems. In 1950, he returned to work for Selden Crary on power systems, where his ideas proved to be most useful, employing the large computers then becoming available. One of his associates wrote two books solidly based on Kron's work that tell how to use computers to control the distribution of electric power in large systems. (8)

Kron spent the years between 1953 and 1963 in laboratory and turbine assignments. In 1963 he joined the Analytical Engineering Division; there he became closely associated with Harvey H. Happ, the editor of this book, who expanded and applied his work to power system problems. Kron retired in 1966 at the age of 65. Kron told me that Happ was his interpreter in electrical engineering, who had his entire confidence. With Kron's encouragement, Happ has written a book that presents the foundations of Diakoptics. (9)

During all the years, as Kron moved around the company, he worked at night and on weekends on many, many ideas, often far removed from his assigned tasks and beyond the understanding of his associates. As his daughter wrote: "His evenings, his free time, were all spent at his desk in the dining room. - Almost every time he went to a movie, he'd impatiently leave the theater part way through the picture, and walk home. - Even though he was in the theater physically, he rarely paid attention to the movie dialogue, and throughout the evening every time the audience reacted, daddy would have to ask mother what was happening. I used to pretend I wasn't related to that curious couple next to me whose whispered summary of the plot in Hungarian punctuated the silence around me." (10)

During these years, he kept up an extensive correspondence with students who asked about his papers and with friends and admirers around the world. Many students said their success in pursuing graduate education was due to Kron's urging. In England and Japan, especially, his ideas were pursued, forming the basis for many papers. On hearing of Kron's death, Dr. Kazuo Kondo wrote: "If I count scientific friends whom I miss for the rest of my life, Kron comes first. For many years I dreamed of meeting him. Whether it would be in the U.S.A. or in Japan, I was uncertain about. He was once expected to visit Japan as a Fulbright scholar. However, to our disappointment it was not realized. In Europe I followed Kron's traces in London and Kent as well as in Liege, where he had been awarded the Montefiore Prize. It was in my plan to find an opportunity to call on him in Schenectady. But, alas! America without him seems to lose its charm to lure me."

Kron was a puzzle to the company executives who were responsible for his assignments. They wanted to tell him what to do, and to put him in a place where his talents would yield immediate, commercially useful results, and to adjust his salary accordingly. This was difficult, because Kron's value was largely in the inspiration he gave to others and in distant objectives that seemed to business managers to be merely dreams. As they tried to make the best use of him, he was shifted hither and yon. In all, he worked for fifteen different managers while with GE. This contrasts with the treatment of Steinmetz, under E.W. Rice. Steinmetz was retained in a staff position and problems were brought to him, instead of moving him to where the new problems were. Thus Steinmetz became known as the "Supreme Court" of the company, and he wrote and lectured freely on whatever topics interested him. Kron had to pursue his own ideas outside of office hours.

This was no real hardship, however, as the company did give Kron complete freedom to publish his ideas, and to maintain contacts with a wide circle of interested men, outside as well as inside the company. Some commercial men, who did not like the engineers to talk about new developments, said that it would do no harm for Kron to write, as no competitor would understand him anyway.

This very point, that his writings were voluminous, and not too clear, was a prime reason for the controversies that always swirled about him. Some who penetrated the depths of his thoughts felt that they could say it more clearly. Others found mistakes in his proofs, or things that he omitted, and many of these published their views. Some belittled Kron, others ignored him or claimed as their own ideas he had developed. Kron felt all these slights keenly. As any one must be who pursues a lonely road, he was sustained by a sense of mission and by an inner conviction that he was rendering a great service to the engineers who would come after him. Some of his critics felt that he was just plain conceited. After all, it is hard to draw a line between sublime confidence that one is right, even though much of the world denies it, and the misguided idea that some foolish idea is important.

Time has shown, however, that Kron was almost always right in his conclusions, eventhough some of his proofs were faulty. He had a remarkable intuition, an unerring sense of the fitness of things prescribed by nature, that led him without fail to the right results. And he had a delightful sense of humor that enabled him to shake off the hard feelings aroused by critics. He was very much like Heaviside in his views about critics. Some one told Heaviside that his writings were too complex. "No," said Heaviside, "perhaps the reader's mind is too simple."

After retirement, Kron continued to work on his theories, with the conviction that he must write several more books, especially on the crystal computer that he thought would be his crowning achievement. Just a few days before the onset of his fatal illness, he told his wife, "My work is so beautiful that I wouldn't give it away for a million dollars." He died, after a short illness, on March 25, 1968.




References and Notes

(1) Gabriel Kron, Philip Alger (ed.),
The Life and Times of Gabriel Kron or Walking Around the World - And Tensors.
Mohawk Development Service, Schenectady, 1969, p.8.
- - To my knowledge the only place to get this book from is
Union College, Schenectady, NY, USA Schaffer Library, keyword=Gabriel Kron.

(2) Kron, Life and Times, pp. 10-16.

(3) After graduating from the University of Michigan, Joseph Kron worked as art engineer for various employers in the United States, until his death in New York in 1966. He was never married.

(4) Kron, Life and Times, p. 74.

(5) Kron, Life and Times, passim.

(6) ibid., p. 135

(7) ibid., p. 161

(8) Leon K. Kirchmayer, Economic Operation of Power Systems. New York: John Wiley & Sons, Inc., 1958.

Leon K. Kirchmayer, Economic Control of Interconnected Sysiems. New York: Wiley.& Sons, Inc., 1959.

(9) Harvey H. Happ, Diakoptics and Networks. New York and London, Academic Press, 1971.

(10) Kron, Life and Times, p. 313.

CONTENTS Foreword EDWIN K. TOLAN VII Introduction H. H. HAPP 1 The evolution of an engineering scientist PHILIP L. ALGER 5 On G. Kron's methods and achievements BANESH HOFFMANN 19 G. Kron and large-scale systems engineering THOMAS J. HIGGINS 37 Kron's contributions to the theory of induction motors PHILIP L. ALGER 43 Damping and synchronizing torques og induction motors CHARLES CONCORDIA 53 Kron's tensor analysis J. W. LYNN 65 The development of diakoptics H. H. HAPP 83 Application of Kron's concepts to the field of system engineering HAROLD CHESTNUT 121 The influence of G. Kron in the United Kingdom ALFRED BRAMELLERAND D. W. MORTIFEE 143 An Oriental expansion of Kron's science beyond electrical engineering KAZUO KONDO 153 APPENDIX Publications of G. Kron 165

INTRODUCTION

H.H.HAPP (1928-), EDITOR
Schenectady,New York

THIS BOOK focuses on the work of Gabriel Kron (1901—1968). It deals with Kron as a person, with his accomplishments, and with his impact on present technology.

Kron was a remarkable individual, a pioneer whose true achievements were appreciated by relatively few when his works were first published but whose methods of systems analysis are now widely employed.

His honors included the Montefiore Prize of the University of Liège, Belgium in 1935; an honorary Master of Science in Electrical Engineering from the University of Michigan in 1936; the Coffin Award of General Electric Company in 1942; an honorary Doctor of Science from the University of Nottingham in 1961; and Patron and Honorary Memberships in the Tensor Society of Great Britain and the Research Association of Applied Geometry of Tokyo, Japan.

Gabriel Kron was a person with an unusual combination of talents who conceived new theories by way of coping with actual physical problems and their practical solutions. He enjoyed and sought out problems that others had unsuccessfully tried to solve, in the hope of filling a gap in the span of scientific knowledge.

For example, his work on specific problems in electric motors and generators led him to pioneer in the use of tensor analysis for application in both rotating machinery and later in static networks. A problem of determining the losses of interconnected power systems led him to conceive a piecewise procedure he called "Diakoptics" for solving large scale systems by tearing; in this approach, the subsystems of the larger system are first solved, and then the solutions of these subsystems are modified to take their interconnection into consideration. Problems in forecasting and optimization, in later years, led him to pursue extensions of his earlier work to include all parameters of the dynamic electromagnetic field by means of the still unexplored polyhedral network theories.

Gabriel Kron and I were close working associates and cooperated on a number of joint projects. I had been familiar with Kron’s work since the early l950s but met him for the first time in 1958. In the ensuing four years, we consulted frequently. In January of 1963, Kron joined the department with which I was associated and we became close working colleagues until his retirement from General Electric in July 1966. We maintained our close working relationship until his death in March 1968.

Kron was a healthy and robust man and credited his physical strength for his capacity to continue his work despite criticism that was sometimes severe. He was a strong believer in physical exercise. He walked home from work nightly, a distance of about four miles. I sometimes accompanied him on these walks, and we usually made much progress on concepts and theories with which we were concerned. He was continually occupied with his work. Work required no self-discipline on his part; rather the opposite: it took force to get him away from his work.

I recall discussing other than scientific matters on our walks home from work. But he could stand "small talk" for only a short period of time, and our conversation had to return to his science, as he called it.

Concepts such as the primitive network, open and closed-path contours, the transformation tensor, among others, never became stale topics to him; in fact, we discussed these concepts for hours at a time over a period of many years.

It was a pleasure working with him. He was a warm and friendly human being, well-liked by almost all who knew him, and a constant source of encouragement and inspiration. His appetite for work, his motivation for progress, his creative imagination, and his tremendous enthusiasm did not abate in the least as the years went by; in fact, he always felt that greater achievements were yet ahead of him.

Kron spent almost his entire career in Schenectady. He loved small-community living and he enjoyed associating with other creative leaders within General Electric in Schenectady, a world focal point and research center in the field of electric power.

The present volume deals with Kron’s life and work under three broad headings - his own career, the technical aspects of his work and some of its applications, and his international influence.

Professor Philip L. Alger sketches Kron’s early life and education, while Professor Banesh Hoffman provides an insight into his working methods, his personal difficulties, and some of the technical problems his work posed for those who attempted to follow it; Professor Thomas J. Higgins outlines the broad scope of Kron’s achievements.

Among the five papers which concern themselves with detailed areas in Kron’s work, the first paper by Professor Alger describes Kron’s contributions to the theory of induction motors, and the second paper by Dr. Charles Concordia, describes Kron’s contributions to practical problems in induction motors. Professor J. W. Lynn deals with tensor analysis and its use by Kron. My own paper describes the problem through which Kron conceived Diakoptics, its theory, and some of its practical applications to systems; and the section ends with an article by Dr. Harold Chestnut on Kron’s influence in the broader field of system engineering.

The third group consists of an article from Britain and one from Japan. The paper from Britain, by Professor A. Brameller and Mr. D. W. Mortifee, describes various engineering applications of Kron’s work and his influence in Britain. Professor Kazuo Kondo’s paper describes the impact of Kron’s work in Japan.

A complete list of Kron’s publications is added to the appendix. By bringing to bear methods from abstract mathematics on broad problems in system engineering and physics, Kron’s genius for generalization and unification made him a controversial figure throughout his career. But Kron knew what he was and what he was not. He never considered himself to be a mathematician; his logic was not mathematical but largely intuitive. Many of his peers in the engineering establishment did not accept the validity Of his unconventional methods, partly because he often failed to prove their validity and partly because they did not recognize the need for such seemingly complex and unproven methods; neither did they recognize the full potential of those methods.

Time has marched on and the methods pioneered by Kron are now widely employed and are taught in almost all universities. A variety of courses in system engineering, netwotk topology and energy conversion, among others, contain material he conceived, although the source is only rarely acknowledged.

Kron’s influence extends far beyond the U.S.A. The Tensor Society of Great Britain came into being to further the understanding and applications of tensor analysis; and he had a strong influence on the activities of the Research Association of Applied Geometry of Tokyo.

We hope that this book not only describes the broad implications of Kron’s work but will also point the way to those individuals who wish to study hjs methods in depth.

As the editor of this book, I express my thanks to the contributors for their papers and to the Director of Union College Press, Bernard R. Carman, for his help in editing all manuscripts and for countless other chores performed in the.publication process.







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FOREWORD

EDWIN K. TOLAN

Librarian of Union College



EDWIN K. TOLAN, a native of Montreal, took his Bachelor of Arts degree at McGill University in 1949 and a Bachelor of Library Science at McGill in 1954. He earned a Master of Arts at the University of Glasgow, Scotland, in 1951, and took his doctorate in the field of philosophy at the UniversitC de Montreal in 1959. He has been Librarian of Union College and Professor of Philosophy at Union since 1962.

Added by this website from a clipping of a local newspaper, date unknown - courtesy of Union College Schaffer Library, Mrs. E. Fladger and J. Spallholz.
Todays's Obituaries - Services for Dr. Edwin K. Tolan, 54, of Schenectady head librarian at Union College in Schenectady, will be at 2 p.m. today from the Baxter Funeral Home in Schenectady. Burial will be in Vale Cementery, Schenectady.
Dr. Tolan died Sunday at his home in 1443 Regent St.
Born in Canada, he had worked as a teller at the Royal Bank of Canada, and served in the Canadian Army during World War II. Dr. Tolan graduated from McGill University in Montreal, the University of Glasgow, and the Institut d'Études Medievales, Université de Montreal.
He began his library career at Hamilton College in Clinton, Oneida County, in 1954. Three years later be became librarian at Washington and Jefferson College in Washington, Pa., a position be held until joining Union College in 1962. Dr. Tolan was instrumental in establishing the Capital District Library Council for Research Resources in 1967. He was a trustee of the council and served as its first president.
Dr. Tolan established the "Union Catalog" containing listing of volumes in eight countries and developed an extensive bibliographic collection. He was a member of the American and state library associations and the Bibliographical Society of America. He was a trustee of the Schenectady Chapter of the National Railway Historical Association and was a member of the friends of the Schenectady Library.
Survivors include his wife, the former Elizabeth MacLennan; and two daughters, Joanne and Jennifer Tolan, all of Schenectady.


THIS BOOK is an outgrowth of the Schaffer Library Symposium, Gabriel Kron, the Man and His Work, held on October 14, 1969 at Union College. The editor has broadened the scope of the collection by adding to the papers first delivered at the Symposium other papers in order to give the reader a greater opportunity to appreciate the richness of Kron’s contributions to modern science and engineering. He has not, however, lost sight of the fact that the initial inspiration and impetus were provided by the Symposium and has therefore thought fit to include my comments on the genesis, purpose and outcome of the Symposium.

It may seem strange that the staff of Schaffer Library took the lead in organizing the Gabriel Kron Symposium, but there were several factors which contributed to this development. Gabriel Kron had used the library resources of Union College for many years and had the respect of the library staff; the late husband of our former Circulation Librarian, Dr. Philip Stanley, coined the term "Diakoptics" at the special request of and for the specific use of Kron; his son-in-law, Dr. Michael Pincus, a Union College alumnus, and Dr. Harvey Happ collaborated in initiating the Gabriel Kron Book Fund for mathematics at Union College; and, finally, there was my own personal friendship with him.
Gabriel Kron was one of those rare library patrons who never take the library for granted. His use of Union College Library facilities was mainly for browsing purposes. When he would find a book or periodical article which could be of use to him in his research, he radiated excitement. I well remember one of his most important discoveries, Samuel I. Goldberg’s Curvature and Homology (New York: Academic Press, 1962). For weeks he told and retold the details of his serendipitous adventure. The fact that he went out and bought a copy for his own use - a copy which he could mark and use without hindering someone else who might want to use the book - was typical of his consideration for others.

Kron gleefully maintained with consistency that he was not a "social lion." This maxim provided him with a convenient dodge and in following it he was able to organize all his free time around his real joy - his research - only permitting himself to be interrupted during, visits by his children and his grandchildren. It is interesting to note, however, that it was at an evening party in Schenectady that Kron met the late Philip Stanley, and sought help in his search for a distinctive name for his then newly-developed method of solving complicated problems in easy stages. Gabe’s personal account of this evening led me to suspect that this encounter gave birth not only to the term "Diakoptics" but also to a consciousness of the rich possibilities of publicly disclaiming any pretensions to being or wishing to be a "social lion."

The response to a general appeal for funds to establish the Gabriel Kron Book Fund was a gratifying one for the family and close friends of Gabriel Kron. I was ill at the time of Gabe’s death but suggested the fund to his son-in-law, Dr. Michael Pincus, who in turn discussed it with Ann Kron and Dr. Harvey Happ. The successful appeal was made by Dr. Happ with the assistance of my secretary, Miss Thora Girke, who attended to the details of seeing that contributions were credited to the proper account, that formal acknowledgments were made, and that proper records were kept. Contributions were received from across the United States, from Great Britain, Australia, Canada, France, and Japan; they were small contributions, only two being $100 or more. Thus, there were many contributors, and most of them were, I am pleased’ to say, from the electrical engineering community. I make this point because during my association with Kron he steadfastly identified himself as a scientist, not an engineer. This was a distinction made by an embittered man who resented the treatment meted out to him by the electrical engineering establishment. Frequent tests, of his reflexes on this point never caught him off guard; his feelings ran too deep. One would argue in vain that pettiness and narrowness were characteristic of the avatars in any circle of intellectual activity; in this matter his fluid ability to generalize deserted him totally. Regardless of the validity of the distinction, it was an important one for him, and this can be seen, I think, in the paradoxical fact that Gabriel Kron was a serene man, well disposed towards others, confident in the validity of his work, and at peace with himself. To identify himself as a scientist was to disassociate himself from institutional animosity, free the spirit for creative work, and permit the mind to express itself with magnanimity. That the Gabriel Kron Book Fund was established for the purpose of purchasing books in mathematics for Schaffer Library is not without some significance.

Gabriel Kron enriched my life, and as I look back I cannot but regard my having bought a house next door to Ann and Gabe Kron when I came to Schenectady in 1962 as anything other than a stroke of great fortune. Only an accident of this kind could have brought together a person of international reputation who had made singular contributions to the solutions of a number of modern scientific and engineering problems, and an obscure librarian who, by reason of background and inclination, held small seminars on medieval philosophy. Our friendship did not grow out of back yard conversations, nor out of mutual interests in specific scientific or philosophical problems; it grew out of a mutual interest in walking - walking as a form of physical exercise and relaxation. There were three distinct phases to this "walking friendship:" daily late afternoon walks, evening walks, and weekend mountain-climbing excursions.



Photo and © Schaffer Library, Union College, Schenectady, NY, USA



Gabriel Kron lived four miles from his work at the General Electric Company, and walked home each evening, while I lived two miles from Schaffer Library and walked to and from work every day. Gabe’s route took him past Schaffer Library at approximately 5 each evening, the time when I would normally be leaving, so without any prearranged appointment we just met and walked home together. Since I could never be sure of leaving on time, nor be sure whether Gabe had passed, I used to watch from the portico of Schaffer Library across the campus to the walk between the Administration Building and Silliman Hall, and became adept at distinguishing him by the rhythm of his gait even during the winter when only his silhouette against a distant street light would be visible. Our conversation usually began with small talk about managers who were profit-product oriented, shop-orders, family affairs, the electrical engineering establishment, Union College, politics, etc. Gabe usually set the tone of these conversations, and I invariably tried to steer it in the direction of his research. I could not, of course, discuss his research in technical terms, conversations in that vein took place only on those occasions when Harvey Happ was with us. I wanted to know the genesis, purpose, use, and potential of his research, and especially to probe the mind of a modern creative scientist. Until I heard Professor Hoffmann’s paper at the Symposium it remained incomprehensible to me that Kron, admittedly not a master of the English language, had been severely criticized for obscurity, for in our conversations he could explain what his research was about in unequivocal language and with a lively and vivid imagery. Sometimes our conversations became extremely animated because my observations and questions were usually directed to matters of philosophical import. For example. I once criticized his concept of time and when, with some bewilderment, he appealed to the authority of Einstein, I merely replied that Einstein should have read the eleventh book of St. Augustine’s Confessions. At this point there. was no further appeal to authority; undaunted, Gabe took off on his own and when we parted in front of my house he said, "Well, that was a good conversation." Rightly or wrongly, I considered myself flattered. On more than one occasion I took out after him on the apparent contradiction of his contentions that, on the one hand, he claimed to be searching to uncover what was hidden in nature, and yet, on the other hand, always claimed to be creative. I never was able to draw him out fully on the question of whether nature was an object of thought or the thought of an object; but he would always say as we parted. "Well, we have had a good conversation."


Whenever I think of Gabe’s vivid imagery there immediately well tip in me recollections of those additional evening walks, usually with other mutual acquaintances. The passing of a satellite, the starry sky above, and the moon in particular were enough to evoke innumerable conjectures about nature. He was sure that nature had provided short-cuts for travel between planets and the scientist’s task was to find those hidden short-cuts. He repeatedly reminded us that he and a friend had an appointment on the moon, to bring in the year 2000 - an appointment made years before Sputnik. As Gabe left me at the end of those long evening walks to go to his study and take up his research I invariably mused about the New Year’s Eve of the ycar 2000, and I suspect that Gabe had convinced himself that nature would spare him until he had finished his research. If my mind’s eye were his, I am sure he would consider his death to have been simple miscalculation.

I was a reluctant, though full-fledged member of the Kron Klimbing Klub. Gabe was nOt Only the founder, but also the spirit of this club, which was dedicated to climbing small mountains within easy reach of Schenectady during the spring, summer and autumn (some bolder types kept on through the winter but they went without Gabe and me). I say I was a reluctant member not because it was forbidden to be too ambitious - indeed, this I liked - but because I found Gabe to be infuriating during such outings. No sooner would we reach the top and eat our lunch than Gabe would start agitating to descend instead of drinking in some of the beautiful panoramas these climbs afforded, panoramas of that nature he was so dedicated to understanding. As his off-key rendition of Italian opera arias boomed through the bush of the lower levels of a mountain I would always muse that h~ was going home to seek inspiration from that artificial crystal that sat on his study desk - das Kristall, in which he believed he found the optical basis for understanding electromagnetic currents, and thus the basis for a mathematics to solve large and complicated system problems. Yet there was no inconsistency in this attitude. For Gabe, beauty did not lie in the visible panorama of nature, but in the proportions of that underlying network which serves as the ordering principle of the universe - Nature. There was streak of Neoplatonism in Gabe which manifested itself in the mystical tendency to see beauty as the product of creative activity emanating from within, and appreciated only by those who turned inward and were able to follow the difficult path laid out by the great thinkers. But the reward came not in a mystical experience, rather in the joy of intellectual discovery.

To characterize ours as a "walking friendship" is somewhat misleading because the expression fails to indicate the quality of this friendship. If mutual trust which permits the disclosure and discussion of private and confidential matters is a proper measure, then ours was a deep friendship. It perhaps is more important, however, to note here the fact that as I became more and more deeply acquainted with Gabe’s trials and tribulations as an engineer and employee of the General Electric Company, with his aspirations and successes as a scientist, with his devotion to and habits of work, with his states of mind and moods which were always a function of progress or stalemate in the movement of his research, and with his private litany of saints and devils, I became aware of a growing conviction that my "walking companion" was a man of genius. I thus found it difficult to consider the successful establishment of the. Gabriel Kron Book Fund a fully adequate acknowledgement of appreciation for the work of an authentic thinker. In using the terminology "authentic thinker" I have precisely in mind the words of Dr. Raymond Klibansky who wrote, "For in the history of ideas we are concerned with those who dare to make a breach in the solid wall of prejudice which at all times dominates human thought, rather than with those who, the way once shown, follow the lead and secure the ground." ('School of Chartres' in: Twelfth-Century Europe and the Foundations of Modern Society, ed. Marshall Clagett, et al. Madison: University of Wisconsin Press, 1961, p.8.) I considered the special privilege of his friendship to have laid upon me the responsibility of being true to a memory which I cherished. It seemed to me that I should try to make it possible for someone to take up and carry on his work which even in all the solemnity of his deathbed he peacefully and characteristically called "beautiful."

It was gratifying to find others of a like mind, though moved from different motives. Here it is appropriate to single out some names. Among the more avid proponents and advocates of a conference on Kron was Dr. Harold Bibber, Professor emeritus of Electrical Engineering at Union College. He hoped that such a meeting might serve to correct some past injustices; this was a significant motive, as Professor Bibber was certainly not one of Gabe’s favorite people because he had been a member of the electrical engineering establishment at the height of Gabe’s struggle with it. Professor Bibber took the lead in engaging the interest of the members of the Electrical Engineering Department at Union College, and its Chairman, Professor Richard Russ, then currently serving as an officer in the Schenectady Chapter of the IEEE. It is a pleasure also to single out the editor of this book, Dr. Harvey Happ, who Gabe had personally told me was the one person to understand fully his Diakoptics, and whose partiality for such a project arose from genuine admiration for Gabriel Kron. In addition, there were members of the Schaffer Library staff who had known Gabe as a library user for many years and who wanted to know more about this quiet, gentle man who derived so much apparent pleasure from books - yes, books on mathematics. From some preliminary talks with these people there developed a program for the Symposium.

The purpose of the Symposium was to gather together persons who had mastered Kron’s work to the point that they could present an evaluation of it which would be intelligible to those not acquainted with his work, and even, if possible, to those, like myself, not even possessing a rudimentary understanding of science and technology. The emphasis was to be on evaluation; there was no intention of providIng a platform for eulogizers or for disparagers. Furthermore, the evaluation was to be directed to Kron’s lifelong work, not his character. It was hoped that a symposium organized in this way might serve not only to correct some past injustices but also some misunderstandings, might eventually stimulate a young scientist to take up where Gabe had had to leave off, and might even serve to stimulate the establishment of a center, either at Schaffer Library or elsewhere, for Kron studies.

Measured by the testimony received from those who attended, the outcome of the Schaffer Library Symposium was a success. Certainly, it far exceeded my most optimistic expectations. This outcome was due, I believe, entir~ly to the caliber of the participants. With the advice and assistance of members of the Electrical Engineering Department at Union College, officers of the Schenectady Chapter of the IEEE, and Dr. Harvey H app, we succeeded in obtaining participants who were so well versed in Kron’s work, its applications and its influence, that they were indeed able to speak imaginatively and intelligently to a very mixed audience and, perhaps more importantly to each other. Can one come closer than Banesh Hoffmann in capturing the enterprising spirit of genius? Does it not become evident from his paper that Kron’s distinction between the scientist and the engineer was not theoretical, but based solely on personality? It was a singular delight to hear this paper read by Banesh Hoffmann who was, in the words of Kron "the one spark that gave me encouragement to continue my research." The pragmatic aspects of Kron’s work came through beautifully. Frankly, I was astonished to learn that his work was now being widely used in architecture, ship building, nuclear power engineering, and even in the electrical power industry for the solution of problems first encountered with the introduction of modern sophisticated generating machines. It was encouraging to learn also that some of his work has become commonplace in engineering textbooks. If I was to have a disappointment it came only afterward with the realization that Kron’s work was not being used in medicine to calculate the suitability of drugs for individual patients, though I have since learned that Gabe himself had received inquiries from cancer research specialists about the possible application of his work in their research. Finally, the tone and character of the papers presented at the Symposium were, for the most part, sober and did not lack in critical content.

As to the long range goals of the Symposium, not much can be said to date. There is no library center to which interested researchers might turn in order to study and further the kind of inquiries which Gabriel Kron so successfully set on foot. To the best of my knowledge there is no one who has yet taken up Kron’s work. where he left off; I mean that work which was proceeding, from his "Generalized Crystal Optics" and was not only to give birth to instruments utilizing "the hierarchy of self-organizing polyhedral waves" but also in the final phase to at least a "multidimensional generalization" of the sciences used by electrical engineers. It is true that exegetical work has been published and is in progress, but does this go beyond securing the ground already won? Has anyone yet grasped the unique universe of Gabriel Kron? This perspective of the physical universe so firmly established in his imagination must be the point of departure for breaking new ground.

I believe that Kron did have a firm and vivid image of the physical universe different from anything hitherto imagined, which served as the touchstone for his research endeavors in the late years of his life. I believe also that the evidence for this is to be found in the introduction to the second impression of his Tensor Analysis of Networks (London: MacDonald, 1965). In these beliefs I may be alone, but an account of the circumstances giving rise to them certainly has a place in this book.

When MacDonald approached Kron about republishing his Tensor Analysis of Networks, Gabe consented on the condition that not one word of the text be changed and that he be permitted to write a new introduction. This presented no difficulties for the publisher, but the new introduction did for Gabe since it was to be the vehicle for poetic justice. The first and most obvious statement to be made in the new introduction was that the book was being reprinted after twenty-six years without a single change in the text; but Gabe wanted to go further and add a general reproof of his alleged plagiarizers. At this point he consulted a number of his friends, including myself, and there was disagreement about the wisdom of these additional points. We spent quite a few long evening walks arguing about an appropriate approach. My position was that if he wished to publish such a reproof then he should not stoop to the pettiness of rhetoric but maintain his dignity and magnanimity. He could do this simply and effectively by elaborating on his statement about the lack of changes in the text and by placing the work in perspective.

It must be remembered that for a period during the 1960s Kron had refused to publish. Then he began to publish again, but very cautiously. Thus he was reluctant to take my advice about the new introduction. During two successive Saturdays of intensive discussion about this matter he told me that he was leaving his scientific colleagues behind and entering a new world, that hisnew discoveries would not be understood unless fully elaborated, and if elaborated would be plagiarized. In any event, he had no intention of tipping his hand until he had fully tested his new theories. As I read the new introduction, I find that he temporized. He did put his work in the perspective of its growth and development and did indicate the direction in which he was moving, but all in great generalities. These generalities may be informative, but if they are not then one would have to fall back on a comparison of the preliminary drafts of the new introduction - if such drafts are still extant.

We know that with the successful development of "Diakoptics" Gabe turned his attention to new and greater challenges, to the comprehension of phenomena of a highly complex nature - multidimensional phenomena. From a philosophical point of view each such phenomenon would be considered complex because each was conceived as constituting a system not only of manifold operations, but of manifold heterogeneous operations. The prototype of this increasingly bold speculation and research was Gabe’s "crystal computer or vest pocket computer," as it was sometimes jokingly referred to, which utilized the simultaneous and heterogeneous activity of a crystal under stimulation of a single electric source. The important steps here seem to have been the perception of an analogy between the optical properties of a crystal under stimulation of a single source of light and the mathematical properties of an electromagnetic field surrounding an electric current, together with the successful use of topology to understand and manipulate them.

I say that the "crystal computer" constituted a prototype because I believe thiswork was successfully accomplished in all its theoretical aspects long before his death, for Gabe showed me what he considered to be proof of the superiority of his new instrument. By this time, however, something else had entered his mind. I suspect that with the theoretical background of his work in developing the "crystal computer" it was not a big jump for the mind of a Kron to begin thinking of the universe itself as a crystal. I also suspect that what he was working on when he died was a general theory applicable to the universe as a whole system. After all the universe is a complex phenomenon in which many heterogeneous activities take place simultaneously. Perhaps Gabe had more than a vague intuition about short-cuts for interplanetary travel.







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PUBLICATIONS OF GABRIEL KRON
1930 - 1968




Just prior to Kron’s retirement in July of 1966, he accumulated together most of his publications. He had two purposes in mind.


1. Compiling a bibliography.
2. Collecting all or most of his papers for the Union College Library. The Librarian of Union College, Dr. E. K. Tolan, kindly arranged to have them bound.


The task of compiling the bibliography was undertaken by the General Electric Company Main Library. I would like to acknowledge the work of Miss Olivia J. Yunker, Specialist-Cataloguing, in the General Electric Company Main Library in Schenectady. Miss Yunker compiled the bibliography in essentially the form shown below {modified slightly by this website}. Starting with a list of Kron’s articles, she verified all items and searched the literature for further publications.

H. H. Happ, editor                      




BOOKS

The application of tensors to the analysis of rotating electrical machinery.
Parts I - XVL. Elementary Engineering treatment.
Reprinted from a serial in the General Electric Review, 38, (1935), 39, (1936), 40, (1937), 41, (1938).
Schenectady, N.Y., General Electric Review, 1st Edition, 1938;
2nd Edition 1942, 208 pp. Parts XVII and XVIII added to 1st Edition.

Tensor analysis of networks.
Wiley, New York; Chapman & Hall, London, 1939;
with a new introduction: MacDonald, London, 1965. 635 pp.

A short course in tensor analysis for electrical engineers.
Wiley, New York; Chapman & Hall, London, 1942. 250 pp. Republished as
Tensors for Circuits.
With a new Introduction and List of Publications of Gabriel Kron.
Dover, New York, 1959.

Equivalent circuits of electric machinery.
Wiley, New York, 1951. With a new Preface: Dover, New York, 1967. 278 pp.

Diakoptics; the piecewise solution of large-scale systems.
MacDonald, London, 1963. 166 pp.


ARTICLES, CORRESPONDENCE, DISCUSSIONS, REPRINTS

G. Kron, Generalized theory of electrical machinery.
AIEE Transactions, 49, 666-683 (1930). Discussion, ibid. pp. 683-685.

G. Kron, Induction motor slot combinations; rules to predetermine crawling vibration, noise and hooks in the speed-torque curve.
AIEE Transactions, 50, 757-767 (1931). Discussion, ibid. pp. 767-768.

G. Kron, Tensor analysis of rotating machinery: the transient performance of asymmetrical machines under unbalanced conditions;
Presented at the January 1933 Winter Convention of AIEE. 30 pp.

G. Kron, Vector theory of circuits involving synchronous machines.
I. H. Summers, discussion by Kron.
AIEE Transactions, 51, 325 - 326 (1932).

G. Kron, Non-Riemannian dynamics of rotating electrical machinery.
J. Math. Phys. 13, 103 - 194 (1934).

G. Kron, Quasi-holonomic dynamical systems.
Physics, 7, 143 - 152 (1936).

G. Kron, Tensor analysis of multielectrode-tube circuits.
AIEE Transactions, Electrical Engineering, 55, 1220 - 1242 (1936).

G. Kron, Analyze tensorielle appliquee a l’Art de l’Ingenieur.
Bulletin de l’Association des Ingenieurs Electriciens sortis de l’lnstitut Electrotechnique Montefiore, no. 9 (September 1936), no. 10 (October 1936), no. 1 (January 1937), no. 2 (February 1937), 56 pp.

G. Kron, Invariant form of the Maxwell-Lorentz field equations for accelerated systems.
J. Appl. Phys. 9, 196 - 208 (1938).

G. Kron, Direct-acting generator voltage regulator.
AIEE Transactions, Electrical Engineering, 59, 149 - 157 (1940).

G. Kron, Equivalent circuit of the capacitor motor.
General Electric Review, 44, 511 - 513 (1941).

G. Kron, Equivalent circuit of the salient-pole synchronous machine.
General Electric Review, 44, 679 - 683 (1941).

G. Kron, C. Concordia, and S. B. Crary, The doubly fed machine.
AIEE Transactions, Electrical Engineering, 61,286 - 289 (1942).

G. Kron, Equivalent circuits for the hunting of electrical machinery.
AIEE Transactions, Electrical Engineering, 61, 290 - 296 (1942). Discussion, p. 458.

G. Kron, Equivalent circuits for oscillating systems and the Riemann- Christoffel curvature tensor.
AIEE Transactions, Electrical Engineering, 62, 25 - 32 (1943). Discussion p. 372.

R. B. Bodine, C. Concordia, and Gabriel Kron,
Self-excited oscillations of capacitor-compensated long-distance transmission systems.
AIEE Transactions, Electrical Engineering, 62, 41 - 44 (1943). Discussion, p. 371.

G. Kron, Steady-state and hunting equivalent circuits of long-distance transmission systems.
General Electric Review, 46, 337 - 342 (1943).

G. Kron, Equivalent circuits to represent the electro-magnetic field equations.
Phys. Rev. 64, 126 - 128 (1943).

G. Kron, Equivalent circuit of the field equations of Maxwell - I.
Institute of Radio Engineers Proceedings, 32, 289 - 299 (1944).

J. R. Whinnery, C. Concordia, W. Ridgway, and Gabriel Kron,
Network analyzer studies of electromagnetic cavity resonators.
Institute of Radio Engineers Proceedings, 32, 360 - 367 (1944).

by G. K. Carter, (companion paper to following item by Kron),
Numerical and network-analyzer solution of the equivalent circuits for the elastic field.
ASME Transactions, Journal of Applied Mechanics, 11, A-162 - A-167 (1944).

G. Kron, Equivalent circuits of the elastic field.
ASME Transactions, Journal of Applied Mechanics, 11, A-149 - A- 161 (1944).

Gabriel Kron and G. K. Carter, Network analyzer solution of the equivalent circuits for elastic structures.
Journal of the Franklin Institute, 238, 443 - 452 (1944).

G. Kron, Tensorial analysis and equivalent circuits of elastic structures.
Journal of the Franklin institute, 238, 399 - 442 (1944).

G. Kron,
Electric circuit models of the Schrödinger equation.
Phys. Rev. 67, 39 - 43 (1945).          Read it here.

G. Kron and G.K. Carter,
A.C. network analyzer study of the Schrödinger equation.
Phys. Rev. 67, 44 - 49 (1945).

G. Kron,
Numerical solution of ordinary and partial differential equations by means of equivalent circuits.
J. Applied Phys. 16, 172 - 186 (1945).          Read that one here.

G. Kron, Equivalent circuits of compressible and incompressible fluid flow fields.
J. Aeronautical Sciences, 12, 221 - 234 (1945).

G. Kron and C. Concordia, Damping and synchronizing torques of power selsyns.
AIEE Transactions, 64, 366 - 371 (1945). Discussion, pp.490 - 491.

G. Kron, Electric circuit models for the vibration spectrum of polyatomic molecules.
J. Chem. Phys. 14, 19 - 31 (1946).

G. Kron and G. K. Carter, Network analyzer tests of equivalent circuits of vibrating polyatomic molecules.
J. Chem. Phys. 14, 32 - 34 (1946).

G. Kron, Equivalent circuit of the primitive rotating machine; modifications of basic circuit adapt it to the analysis of many types of machines.
General Electric Review, 49, 43 - 49 (1946).

G. Kron, Equivalent circuits for the numerical solution of the critical speeds of flexible shafts.
ASME Transactions, Journal of Applied Mechanics, 13, A-109 - A-116 (1946).

G. Kron, Equivalent circuit of the primitive rotating machine.
AIEE Transactions, Electrical Engineering, 66, 369 - 372 (1947).

G. Kron, Tensorial analysis and equivalent circuit of a variable - ratio frequency changer.
AIEE Transactions, 66, 1503 - 1506 (1947).

G. Kron, Equivalent circuit of the primitive rotating machine with asymmetrical stator and rotor.
AIEE Transactions, 66, 17 - 23(1947).

G. Kron, F. I. Maginniss and H. A. Peterson Part I - Differential analyzer representation.
Institute of Radio Engineers Proceedings, 36, 70 - 73 (1948).

G. Kron, Tensorial analysis of control systems.
ASME Transactions, Journal of Applied Mechanics, 15, 107 - 124 (1948).

G. Kron, Electric circuit models of partial differential equations.
AIEE Transactions, Electrical Engineering, 67, 672 - 684 (1948).

G. Kron, Steady-state equivalent circuits of synchronous and induction machines.
AIEE Transactions, 67, 175 - 181 (1948).

G. Kron, Stationary networks and transmission lines along uniformly rotating reference frames.
AIEE Transactions, 68, 690 - 696 (1949).

G. Kron, Tensors as organizations.
Matrix and Tensor Quarterly, 1, 3 - 4 (1950).

G. Kron, Classification of the reference frames of a synchronous machine.
AIEE Transactions, 69, 720 - 727 (1950).

G. Kron, Equivalent circuits of the shaded-pole motor with space harmonics.
AIEE Transactions, 69, 735 - 741 (1950).

G. Kron, Tensorial analysis of integrated transmission systems.
Matrix and Tensor Quarterly, 1, 3 - 6 (1951).

G. Kron, Tensorial analysis of integrated transmission systems, Part I, The six basic reference frames.
AIEE Transactions, 70, 1239 - 1246 (1951). Discussion, pp. 1246 - 1248.

G. Kron, Electrical engineering problems and topology.
Matrix and Tensor Quarterly, 2, 2 - 4 (1951).

G. Kron, Tensorial analysis of integrated transmission systems, Part II, Off-nominal turn ratios.
AIEE Transactions, 71, 505 - 512(1952). Discussion, p. 512.

G. Kron, So you are going to study tensors!
Matrix and Tensor Quarterly, 2, 3 - 6 (1952).

G. Kron, A new theory of hunting.
AIEE Transactions, 71, 859 - 863 (1952). Discussion, p. 865.

G. Kron, Tensorial analysis of integrated transmission systcms, Part III, The 'primitive' division.
AIEE Transactions, 71, 814 - 821 (1952). Discussion, p. 821.

G. Kron, Solving extremely large and complicated physical systems in easy stages.
Matrix and Tensor Quarterly, 3, 2 - 4 (1953).

G. Kron, A set of principles to interconnect the solutions of physical systems.
J. Appl. Phys. 24, 965 - 980 (1953).

G. Kron, Tensorial analysis of integrated transmission systems, Part IV, The interconnection of transmission systems.
AIEE Transactions, 72, 827 - 838 (1953). Discussion, p. 839.

G. Kron, A method of solving very large physical systems in easy stages.
Institute of Radio Engineers Proceedings, 42, 680 - 686 (1954).

G. Kron, Tearing and interconnecting: a new type of transformation.
Matrix and Tensor Quarterly, 4, 2 - 5 (1954).

G. Kron, Electric circuit models of the nuclear reactor.
AIEE Transactions, 73, 259 - 265 (1954).

G. Kron, Regulating system for dynamoelectric machines.
U.S. Patent Office, no. 2, 692, 967 (October 26, 1954).

G. Kron, A super-regulator cancelling the transient reactance of synchronous machines.
Matrix and Tensor Quarterly, 5, 71 - 75 (1955).

G. Kron, A physical interpretation of the Riemann-Christoffel curvature tensor; the distribution of damping and synchronizing torques in oscillating transmission systems.
Tensor (new series), 4, 150 - 172 (1955).

G. Kron, Inverting a 256 x 256 matrix; solution of an engineering system by method of sections using a card programme calculator.
Engineering, 178, 309 - 312 (1955).

G. Kron, Detailed example of interconnecting piece-wise solutions.
J. Franklin Institute, 259, 307 - 333 (1955).

G. Kron, Solving highly complex elastic structures in easy stages.
ASME Transactions, Journal of Applied Mechanics, 77, 235 - 244 (1955).

G. Kron, The analytical solution of complex physical structures.
Matrix and Tensor Quarterly, 5, 115 - 120 (1955).

G. Kron, Tearing and interconnecting as a form of transformation.
Quarterly of Applied Mathematics, 13, 147 - 159 (1955).

G. Kron, Simplified form of the factorized inverse matrix.
Matrix and Tensor Quarterly, 6, 2 - 5 (1955).

G. Kron, Recapitulation of the geometrical aspects of Gabriel Kron’s non- Riemannian electrodynamics.
Memoirs of the Unifying Study of the Basic Problems in Engineering Sciences by means of Geometry, 1, 185 - 239 (1955).

G. Kron, Solution of complex nonlinear plastic structures by the method of tearing.
J. of Aeronautical Sciences, 23, 575 - 562 (1956).

G. Kron, How to use the A.C. network analyzer for 'tearing'.
Matrix and Tensor Quarterly, 6, 131 - 134 (1956).

G. Kron, Improved procedure for interconnecting piece-wise solutions.
J. Franklin Institute, 262, 385 - 392 (1956).

G. Kron, Multiple substitution of basic vectors in linear programming.
Matrix and Tensor Quarterly, 7, 1-11 (1956).

G. Kron, Multiple substitution of basic vectors in linear programming.
Matrix and Tensor Quarterly, 7, 48 - 50 (1956). - continued from September 1956.

G. Kron, Electrical power engineering as a spearhead of 'universal' engineering.
Bulletin of Electrical Engineering Education (December 1956).

G. Kron, Diakoptics - a gateway into universal engineering.
Electrical Journal (London), 157, 1940 - 1945 (1956).

G. Kron, Diakoptics - A piecewise solution of large-scale systems. A serial.
Electrical Journal (London), 158-162, a serial of 20 chapters, from June 7, 1957 to February 13, 1959.

G. Kron, A very simple example of piecewise solution.
Matrix and Tensor Quarterly, 8, 13 - 15 (1957).

G. Kron, Factorized inverse of partitioned matrices.
Matrix and Tensor Quarterly, 8, 39 - 41 (1957).

G. Kron, Tearing, tensors and topological models.
American Scientist, 45, 401 - 413 (1957).

G. Kron, Numerical example for interconnecting piece-wise solutions of elastic structures.
RAAG Memoirs, II, 389-403 (1958).

G. Kron, Diakoptics - the science of tearing, tensors and topological models.
RAAG Memoirs, II, 343 - 368 (1958).

G. Kron, A generalization of the calculus of finite differences to non-uniformly spaced variables.
AIEE Transactions, 77, 539 - 544 (1958).

G. Kron, Multi-dimensional space filters.
Matrix and Tensor Quarterly, 9, 40 - 43 (1958).

G. Kron, Basic concepts of multi-dimensional space filters.
AIEE Transactions, 78, 554 - 561 (1959).

G. Kron, Self-organizing, dynamo-type automata.
Matrix and Tensor Quarterly, 11, 42 - 52 (1960).

G. Kron, Graphs as illegitimate models of electrical networks.
Matrix and Tensor Quarterly, 12, 1 - 10 (1961).

G. Kron, Elastic structures from the point of view of topological network theory.
RAAG Memoirs, III, 23 - 32 (1962).

G. Kron, Power-system type self-organizing automata.
RAAG Memoirs, III, 392 - 417 (1962).

G. Kron, Camouflaging electrical l-networks as graphs.
Quarterly of Applied Mathematics, 10, 161 - 174 (1962).

G. Kron, Multi-dimensional curve-fitting with self-organizing automata.
Journal of Mathematical Analysis and Applications, 5, 46 - 49 (1962).

G. Kron, The misapplication of graph theory to electrical networks.
AIEE Transactions, 81, 257 - 267 (1962).

G. Kron, The frustrating search for a geometrical model of electro-dynamic networks.
Tensor (new series), 13, 111 - 128 (1963).

G. Kron, Invisible satellites of electric networks.
(Correspondence) Matrix and Tensor Quarterly, 14, 90 - 94 (1964).

G. Kron, Non-linear diakoptics and the optimization of dynamic systems.
(Correspondence) Matrix and Tensor Quarterly, 14, 127 - 129 (1964).

G. Kron, Invisible dual (n-1)-networks induced by electric l-networks.
IEEE Transactions, CT-12, 464-470 (1965).

G. Kron, Tensorial and topological foundations of electric networks.
In: Perspectives in Geometry and Relativity; essays in honor of Vaclav Hlavaty.
edited by Banesh Hoffmann (University Press, Illinois,Indiana, 1966), 491 pp. pp. 217 - 229.

H. H. Happ, discussion by G. Kron, Orthogonal networks.
IEEE Transactions, PAS 85, 291 - 293 (1966).

H. H. Happ, discussion by G. Kron, Special cases of orthogonal networks - Tree & link.
IEEE Transactions, vol. PAS 85, 889 - 890 (1966).

H. H. Happ, discussion by G. Kron, Z Diakoptics - Torn subdivisions radially attached.
IEEE Transactions, PAS 86, 766 - 767 (1967).

H. H. Happ, discussion by G. Kron, Special cases of orthogonal networks - Mesh & nodal networks.
IEEE Transactions, PAS 87, 62 - 63 (1968).

G. Kron, Four abstract reference frames of an electric network.
IEEE Transactions, PAS-87, 815 - 824 (1968).

G. Kron, Non-Riemanian Dynamics of Stationary Electric Networks.
RAAG Memoirs, IV, 358 - 372 (1968).