Miloš Radojčić

DIGITIZATION OF THE WORKS OF DR MILOŠ RADOJČIĆ http://elib.mi.sanu.ac.rs/files/journals/ncd/16/ncd16025.pdf

Abstract. We present the digitized works of dr Miloš Radojčić, professor at the University of Belgrade and corresponding member of the Serbian Academy of Science and Arts. These works have been digitalized at the Mathematical Institute of the Srebian Academy of Science and Arts and will be exhibited as a part of the virtual library of the National Center for Digitization. In the background of all of Radojčić’s scientific work was the teaching of anthroposophy, which led him to, at the time, most contemporary problems in mathematics and mathematical physics. 1. Introducation For a couple of years the virtual library of digitized mathematical papers related to our country is being developed under the auspice of the National Center for Digitization. This paper will present the digitized works of dr Miloš Radojčić (1903–1975), professor at the University of Belgrade and corresponding member of the Academy of Science and Arts. We have also pointed to the place dr Radojčić holds both in our and the world’s science. In the background of all of Radojčić’s scientific work was the teaching of anthroposophy and in it geometry. Anthroposophy led him to, at the time, most contemporary problems in mathematics and mathematical physics. The digitization of professor Radojčić’s works was done at the Mathematical Institute of the Serbian Academy of Sciences and Arts and will be exhibited as a part of the virtual library of the National Center for Digitization [13]. 2. Life and Scientific Work Miloš Radojčić belongs to the group of our scientists of which little has been and is written about. He passed unnoticed, almost invisible through our science, but only as a man – not as a scientist. His work, although still not fully evaluated, bears a mark of an ingenious personality with broad and well-founded interests ranging from mathematics and physics to philosophy, religion and art.

Based on some of his qualities, he would be one of those intellectuals between two  wars, actually in the first phase of that period, who appreciated and nourished not only their  field of study but also the general culture. That circle of people brought a new spirit to our  milieu, the spirit that was later pushed out by narrow specialists. The broadness of the view  made those people appreciate not only their own science but other values as well, and made  them look at their own results with modesty, with more doubts about their real significance.  This broadness is reflected in Radojčić as an artistic talent that he exhibited as he engaged in  music as an amateur, in art almost as a professional, in poetry but also in philosophy and  science, since everything he did was imbued with threads of an artistic soul, refined to the  utmost sensitivity. 26  Miloš Milovanović

Radojčić was born in Zemun, on 31 August 1903. Four grades of, at the time,  elementary school he completed in Zemun,  where he also started gymnasium but only  completed the first grade. During the First World War he received his education in France and  Switzerland, where he completed the second through the sixth grade of gymnasium, and the  seventh and the eighth grade along with the final examination he completed in 1921 in the  town of his birth.  

The same year he enrolls at the Department of Mechanical Engineering at the  Technical Faculty in Gratz , but abandons it the next year, disappointed by the pragmatism of  the instruction, that some professors emphasized in particular and with pride. His outstanding  artistic soul, especially in fine arts, was looking for a profession imbued with art. For this  reason he enrolls at the architecture at the Technical Faculty in Belgrade, but stays there for  only one semester; because Radojčić perceived art as a road to knowledge, and he did not find  that studying architecture. In 1923 he transfers to the Mathematical group at the Faculty of  Philosophy in Belgrade where he graduated in 1925, since he was able to get credit for the  work done while studying technique. In 1928 he got his PhD from the Faculty of Philosophy  for the doctoral thesis “Analytical functions expressed in terms of convergent series of  algebraic functions”.  

Professor Radojčić’s scientific activity deals for the most part with the theory of  analytical functions and can be classified into three thematic circles. The first one Radojčić himself defined as expressing general multiform analytical functions, on any type of their  domain in terms of convergent series of algebraic functions. At the  very beginning of his  career, in his PhD thesis Radojčić gave his generalization of the well-known theorems of  Weirstrass and Runge for the case of an analytical function on  an arbitrary region of the  corresponding Riemann’s surface. Radojčić added on this result and improved it on many  levels in his thesis and in a series of later articles published before and during the war. The  final form of the result, which among other things contains the theorem stating that any  analytical function on any region of its Riemann’s surface can be uniformly approximated by  a series of algebraic functions, can be considered as a maximum possible generalization of the  above mentioned theorems of Weierstrass and Runge. It is important to say that those results  of Radojčić represented for the long period of time the most important achievements in its  field. It was only twenty years after the publishing of Radojčić’s thesis that the German  mathematician Helene Florak produced the results comparable by the importance and  difficulty to those of Radojčić; however, her result only nicely complement those of Radojčić and cannot be, by any means, considered their substitute. 

The second thematic circle dealt with the problem of dividing Riemann’s surface into  leaves, which was one of the basic problems of the geometric theory of functions during one  period of its development. Radojčić proved many central theorems related to these complex  problems and introduced a general approach to the process of dividing any Riemann’s surface  into leaves for the case of unbounded Riemann’s  surface, when it is possible to talk about  such a division in the usual sense. According to the competent  opinion of the German  mathematician E. Urlich, those theorems achieved the maximum possible results within the  realm of the applied method. These and other results, published for the most part in the  journals of our Academy and University, did not receive the needed publicity, which in turn  enabled the appropriate response of the international mathematical audience to the similar  results obtained by almost identical method by Japanese mathematician Schimicu , who  published his papers in the world known journal. Upon closer examination of those papers,  however, one can form an impression that the priority should be on professor’s Radojčić side,  both in time and essence. Both Radojčić and Shimizu used those theorems as a base for their Miloš Milovanović 27 further studies of automorphous functions. According to one of the major results of Radojčić,  obtained during further investigations, every meromorphous function is in a certain sense  automorphous. 

The third group of his papers deals with geometrical and topological properties of  analytical function in the vicinity of essential singularities, with the special consideration of,  so called, problem of the type of the Riemann’s surface. This is about stating the criteria so  that Riemann’s surface is of elliptical, parabolic or hyperbolic type. Here, among other things,  Radojčić offered two variants of conditions sufficient for Riemann’s surface to be of parabolic  or hyperbolic type.  

All of these Radojčić’s papers belong to the geometric theory of analytical functions.  In relations to this, he can be considered an independent pioneer of certain methodological  concepts of the theory of analytical functions. Also, it can be said that certain geometrism, a  subtle and fluid geometric spirit and thinking style, represents an important component and a  deep dimension of both internal unity and continuity of professor Radojčić’s scientific work  and his overall figure as a mathematician. Actually, this spirit represents that life-forming  center, whose fertile and inspiring glow permeates all areas of his mathematical work,  including his invaluable contributions to the introduction, establishment and design of all  forms of contemporary instruction of geometry at the University of Belgrade.  

In 1938, after being promoted to Assistant Professor at the Department of  Mathematics of the Faculty of Philosophy, Radojčić, parallel to lecturing in the theory of  analytical functions, took upon himself the task of establishing the first course of synthetic  geometry, entitled Elementary geometry.  In 1945, when the new curriculum was made,  Radojčić proposed the introduction of two new geometric disciplines: descriptive and higher  geometry. Such an ambitious plan could not be carried out that easily with an insufficient  number of teachers. Radojčić took upon himself obligation to teach three geometric  disciplines: descriptive, elementary and higher geometry. 

The course Elementary geometry was not only the first systematic course of synthetic  geometry, but also the first axiomatic course ever taught at the University of Belgrade. In his  lectures, professor Radojčić started to develop a rigorous approach to defining concepts and  proving theorems, an approach never previously  applied at the University of Belgrade, that  called for proving even the most obvious assertions not included in axioms. In a certain way,  Radojčić presented in this course his own axiomatic approach to the Euclidean geometry.  Being of the opinion that the deductive method was more evident when the number of starting  concepts was minimum possible,  his starting concepts were the  point and relations  between and congruent. Since he defined the line and the plane using the order of points, in Radojčić’s  work axioms of order preceded axioms of belonging. In his book, based on this course,  Elementary geometry, he pointed out that this approach was consistent with the using of set  theory in building geometry, where the line and the plane are seen as sets of points, not as  separate elements of space.  

Radojčić’s work on axiomatically establishing the special theory of relativity is also of  great importance for the deeper understanding of his relationship with geometry. He  considered Einstein’s relativistic physics to  be a return of geometry to the source of the  experience of which it had descended long ago and the once again established lost link  between physics and geometry; the link that showed that actually geometry was a branch of  mathematical physics. In his discussions  of geometry, he named this approach  internal  viewpoint,  since it derived geometry from the immediate internal experience of physical  reality, to which we also belong [5]. 28  Miloš Milovanović

In the list of Radojčić’s scientific works eight units are related to this area. The  summary of his work in this area is presented in the monograph  Une construction  axiomatique de la théorie de l’espace-temps de la Relativité Restreinte, published in the  Special editions of the Academy of Arts and Science in 1973. Since he published the first  paper in 1933 and the final monograph in 1973, it is evident that his interest in the theory of  relativity lasted for full forty years.  

Axiomatic establishing of the space-time  continuum of the theory of relativity  interested mathematicians since the beginnings of its existence. In the papers that followed,  some authors accept for the most part already formed structures and apply them to the topic of  their interest, resulting in a short paper  and relatively small number of axioms. Radojčić,  however, believes that the topic  of such fundamental and elementary importance, such as  kinematics of the theory of relativity, that in its way involves elementary geometry as well,  deserves an independent and in the contemporary sense, elementary treatment. In this way, he  starts with axioms that are, in their physical interpretation, closest possible to the noticeable  facts, avoiding to use even the analogy with the axiomatic system of the elementary geometry. 

Guided by his own language minimalism, Radojčić chooses for his basic concepts  signals or flashes of light, named instantaneous events and points they originated from or can  be seen at, named  material points. Along with these two concepts, there are three basic  relations: to happen, to be seen and before. The whole theory is based on five basic concepts  and 27 axioms classified  into nine groups. Radojčić derives Lorentz’s transformations for  light-metric bodies as a logical consequence of basic geometric properties of the spread of the  light. In this way, Lorentz’s  transformations are totally independent from the physical  experiment, to the point that they cannot be even overruled by the experiment. Experiments  are required only to establish whether solid bodies, which provide on Earth the base for all  measurements, even cosmical events, have properties of light-metric bodies. The affirmative  answer to this question, stating that solid  bodies are indeed systems of light-metric bodies,  already lies in the fact that every spectral line measured in the matter of solid state of  aggregation, has the constant wave length. 

The work of professor Radojčić was neither the first nor the only breakthrough made  by our scientists into the geometric kernel of the theory of relativity. Already in 1910,  Vladimir Varićak, professor at the University of  Zagreb, highly versed in non-Euclidean  geometry, gave an interpretation of the special theory of relativity in the geometry of  Lobachevski [12]. The following year, in 1911 he reaches a completely new understanding of  the connection between those two areas of mathematics and proves that, just as the Newtonian  kinematics is derived from the Euclidean geometry, so can the basic theorems of Einstein’s  non-Newtonian kinematics be derived from the geometry of Lobachevski. The work of  professor Radojčić, along with papers of Vladimir Varicak represented our rare contribution  to this new science. 

In 1959 professor Radojčić leaves the country. It is speculated that the reason lies in  his inability to withstand the socio-political reality of the time he lived in, the reality that  inundated also the University of Belgrade. He used the UN announcement about the  scientific-technical aid to the undeveloped countries of Africa and Asia and from 1959 until  1964 worked as a professor at the University  of Khartoum, Sudan. After that he made the  final move to the National Center for Scientific Research in Paris, residing in a small town  Thonon-les-Bains near the Swiss border. There he died on May 14 th , 1975.  

With the exception of scientific papers and text books, everything that Miloš Radojčić published appeared before the German occupation of Yugoslavia. Nevertheless, his Miloš Milovanović 29 unpublished documents still exist in Goetheanum 1

, in the Swiss town of Dornach, near Basel. Some tens of thousands of pages on cosmology, biology, philosophy, poetry, history, religion, literary interpretations and literary translations and studies and more all wait to be processed and, in a meaningful way, offered to readers. In his written legacy there are no papers related to mathematics [2]. 3. Teaching of Anthroposophy Anthroposophy is a mystic doctrine based on the teaching of Austrian philosopher, scientist and artist Rudolf Steiner. Steiner was originally the secretary general of the Theosophist society 2

for Germany, but he felt that the theosophist teaching was too much  under the influence of Hinduism and Buddhism. Therefore, he felt a need to reveal a spiritual  path in the authentic tradition of the West, an endeavor he carried on until his death.  He was looking for a spiritual understanding of philosophy that enabled perceiving the world  and the history of mankind as a consequence of the influence of higher forces – spiritual  beings, which led the mankind step by step on the evolution ladder. During his life he wrote  more than twenty books and gave great number of lectures, resulting in over 360 volumes of  his collected works.  

Professor Radojčić was an anthroposophist. In anthroposophy he saw the aspiration of  his time to the conscious revival of the primordial spirit of life in the midst of all-devouring  desert of the materialistic view of the world. In particular, he emhasized the aim of  anthroposophy to conduct its research in a spirit that permeated the contemporary culture i.e.  the scientific spirit. What is important for characterization of the spirit of contemporary  culture is less the state of religion, even arts and more the state of the science. The force of  that spirit was most completely realized in the contemporary physics and it is not a  coincidence that the appearance of anthroposophy coincided with the rapid advances in  physics at the beginning of the 20 th

century. This sets anthroposophy apart from both a  dreamy utopism and non-critical acceptance of the given truths, both being an expression of a  way a spirit is enslaved by a soul’s lower forms. 

 Among numerous anthroposophist papers on various topics a special place holds the  paper On our medieval painting’s magical world, published in installments in the magazine  National defense in 1940. The text reveals a thinker of an extraordinary spiritual courage  ready to guide the reader all the way to the  frontier where individual consciousness touches  horizons of all mankind and all times, by journeying deep inside himself in search for the  secret of the time long gone. Without doubt, this text is one of the best studies of medieval  painting ever written. 

Over the centuries, not only do the external circumstances of life change, but also the  world inside a man and that change is deeper and more dramatic than ever thought in our  time. From one century to the other, not only do customs, beliefs and all that enters the soul  from the outside change but also the soul’s  abilities and spiritual powers. Thus, we are 

1 Goetheanum in Dornach serves as a world center of Anthroposophist society and the main center of The School of the spiritual science. The building, based on the project by Rudolf Steiner, was erected between 1925 and 1928 and was named in honor of the famous German poet and precursor of the anthroposophist movement. 2

Theosophist society was founded in 1875 aiming to promote the thinking system developed in the papers of  Helena Blavacka. Theosophy claims that all religions are attempts to assist the mankind in reaching the higher  perfection by “spiritual hierarchy” and consequently, that every religion contains a part of the truth. Practically,  this religious syncretism is mainly characterized by meditative techniques of India and Far East, through which it  made a far-reaching impact on world culture by spreading them even in the countries they came from and where  they were far from being widely used. 30  Miloš Milovanović separated from the Middle Ages by a very concrete spiritual gap, which prevents us from  comprehending the medieval art in the contemporary cultural and civilization context.   What roams around in the consciousness of today’s painter, ”problems” that  preoccupy him, did not even exist in the soul of a medieval painter of ours and vice  versa: what bothered a painter of that time during the sleepless nights, spent praying  or lights of his vision that illuminated him throughout his earthly life, all that,  naturally, does not exist for the contemporary painter. It has to be said: in spite of us  standing today face to face with a new, powerful wave of spirituality, such a form of  soul and spiritual life that once existed will never again provide a foundation of the  art of painting; because mankind moves forward, through deaths and resurrections,  and future spirituality will never be the same as the one that passed.   While becoming absorbed in the mood that our old painting awakes, Radojčić notices  the sentiment close to the feeling of an evening, a sunset, an evening twilight, even a night.  Already with first impressions that our old paintings provoke, we are overcome by a  feeling of twilight; perhaps not because paintings tarnished of the century long fume  of wax candles and icon lamp, but because it is really twilight that is being painted.  All that is happening, happens at twilight, even when the painter did not have the  evening on his mind, but a day or a night. On those paintings there is no real day to be  found: in every day there is a mysterious presence of night  or an evening day. And  night is not that dark, but filled with visions of one unearthly day.  The answer to the question why is this, he finds in souls of painters of that time. While  dealing with the general progress of the mankind, he points out that the ancient human  consciousness bears high resemblance to the world of dreams. It was that the human soul was  filled with sights that spoke instead of logical thoughts and that was one all-encompassing  primordial clairvoyance. Spiritual depths or  heights of the universe, reachable neither by  senses nor plain thinking, were within reach of such clairvoyance. Over many a century that  clairvoyance slowly faded and its innate place was overtaken by a freedom, which logical,  critical thought brought along.  Various ancient religious texts foresee this eclipse of  clairvoyant powers and describe it as a night, through which the mankind would have to pass.  It is that night that Radojčić recognizes as the appearance of materialism at the beginning of  Modern times. Using this night as a point of reference, the late Middle Ages represents hours  of an evening twilight. Those are the hours of the ancient clairvoyance’s twilight, in the last  glistening of colors prior to total darkness.   Radojčić then writes about one strange occurrence – days and nights befriend each  other in the same painting, as it is often seen on our old frescoes. The Earth emits light from  its strange rocks and objects on it and the sky is dark blue, like during the night. This is  because old painters aspired not to paint the external world, but to pour the depths of their  soul into paintings. The earthly world with its external light eclipses the splendor of the