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The Life and Work of János Bolyai
György Alexits presented the lecture The Life and Work of János Bolyai at the meeting held on 14 December 1952 of the ceremonial session of the Hungarian Academy of Sciences. It was published, in Hungarian, in the Proceedings of the Department of Mathematical and Physical Sciences of the Hungarian Academy of Sciences 3 (2) (1953), 131-150. On the following day, 14 December 1952, he gave the lecture The Influence of Bolyai-Lobachevsky Geometry on the Development of Geometry to the Academy. We give below translations of the Abstracts of both lectures and an English translation of the first of these two talks. We have used both the Hungarian version and a Russian version of this article in our translation. We have added a few notes. We note that, as well as giving a very good account of Bolyai's life and work, Alexits also give much information about his own political views in the article.
1. Abstract of The Life and Work of János Bolyai.
János Bolyai, one of the greatest mathematicians of all time, was a brilliant genius with a tragic fate. His fate is not just an individual tragedy. It also reflects the struggle of all progressive thinkers of the Hungarian people who dared to stand up for the truth and therefore clashed with the perception of the ruling class. Bolyai fought for the enforcement of a scientific idea, and this idea was revolutionary. The movement of a scientific discipline is not an isolated phenomenon. Its explosive progress is prepared not only by the internal development of the discipline in question, but also by the many thousands of minor and major changes in the structure of society, which are absorbed into the scientific view through the complicated capillary system of life, to finally revolutionise science through the work of a brilliant creative genius. And the revolution, even if it takes place in the field of abstract science, arouses the hatred of the dying ruling class.
2. Abstract of The Influence of Bolyai-Lobachevsky Geometry on the Development of Geometry.
I would like to discuss the impact of János Bolyai's discovery on the development of geometry from the following aspects: first, we will describe those investigations that methodologically follow János Bolyai's synthetic discussion. The results of these investigations lead to the modern axiomatic foundation of geometry. Second, we will outline those investigations that led to János Bolyai's results from other aspects. However, starting from these aspects, geometries that are significantly more general than János Bolyai's discovery were reached. One of the new aspects leads to the group theoretical foundation of geometries through the Cayley-Klein projective approach, while the other aspect starts from Riemann's famous habilitation lecture and ultimately leads to the differential geometric foundation of various geometric disciplines. It is known that the investigations of the forerunners of non-Euclidean geometry, G Saccheri (1667-1733), J H Lambert (1728-1777), and A M Legendre (1752-1833), followed the synthetic-axiomatic method of Euclidean geometry. However, these investigations only revolved around the parallelism axiom, and their goal was to demonstrate that this Euclidean axiom follows from the other axioms. Bolyai and Lobachevsky already purposefully started from the non-Euclidean hyperbolic parallelism axiom, which was the opposite of the Euclidean axiom, and tried to build a system that would give a geometry that was equally possible as Euclidean geometry.
3. The Life and Work of János Bolyai.
János Bolyai was born on 15 December 1802, in Kolozsvár (now Cluj-Napoca), in the home of his mother's father, the surgeon József Benkö. Until the age of two, he was raised in Domáld (in Hungary at the time, now in Mureș County, Romania), on the estate of his father, Farkas Bolyai. In 1804, the family moved to Marosvásárhely (then in Hungary but now known as Târgu Mureș in Romania), where Farkas Bolyai was invited to teach mathematics, physics, and chemistry at the Reformed College.
From a young age, János showed an extraordinary aptitude for mathematics. By the age of four, he already knew the concepts of a circle, radius and centre, ellipse, and even the concept of "sine" was not foreign to him. At the age of nine, his father introduced him to systematic study. At that time, he studied Euclid's books and Euler's algebra. He hardly needed to be taught mathematics, because he immediately grasped every theorem he heard and foresaw its proof. "It was like the devil jumping in front of me," his father said, and he demanded to continue. By the age of 13, he already had skills in infinitesimal calculus. He passed the matriculation exam, the so-called "rigorosum", in 1817 at the Marosvásárhely college (Marosvásárhely is now known as Târgu Mureș), and thus gained the right to continue his studies at the university.
Farkas Bolyai wanted his son to continue his studies of mathematics in Göttingen under the guidance of the "princeps mathematicorum" Gauss. Given that Farkas had been friendly with Gauss in his youth, he wrote him a letter asking him to accept his son János into his home.
However, over time, this relationship cooled, and Farkas received no response to his letter, written in an overly frank tone. Thus, any opportunity for János to achieve a thorough, higher-level mathematical education ceased.
At that time, studying higher mathematics in Hungary was impossible. By the age of fifteen, János Bolyai had mastered mathematics to a level unsurpassed within the country. He could only receive serious training in Göttingen or Paris. But Farkas lacked the funds for this, as his annual salary was 400 forints, not counting payments in kind, which allowed him and his family to live very modestly in Marosvásárhely. Only financial support from aristocratic landowners would have made it possible to educate his son abroad, and this, according to the customs of the time, the popular Farkas could count on. But if someone is a patron of the arts, they also choose the cause they support with their money. And at that time, the conservative Transylvanian aristocracy had little interest in mathematics and was primarily interested in the politics of the Habsburg dynasty. Therefore, Farkas could not hope that the landed aristocrats would finance his son's studies in Göttingen or Paris, but he could hope that they would finance his studies at the Academy of Military Engineers in Vienna, as they had a political interest in this. The landed aristocrats, loyal to the Habsburg dynasty, even considered it their duty to promote the sons of minor nobles to become officers in the Austrian army, in order to weaken the ranks of rebellious nobles by raising their sons and relatives as Janissaries in Vienna loyal to the emperor. A year later, Farkas managed to persuade several aristocrats to cover the costs of János's education in Vienna, amounting to approximately 8,000 forints over four years. And so, in 1818, after successfully passing the entrance exam, János Bolyai was admitted to the Vienna Academy of Military Engineers.
According to the academy's curriculum, János had to study mathematics for another three years; in his fourth year, he studied military subjects. The spirit of the academy corresponded to the views of the Austrian military establishment of the time: students were indoctrinated in the official worldview. Furthermore, the academy trained military engineers, not mathematical researchers, so in the field of mathematics, only the knowledge necessary for military engineers of the time was required. Thus, János Bolyai did not receive the mathematical training on which a future researcher could rely. He was also unfamiliar with the way he dealt with representatives of science, since the closed educational method of the military institution did not encourage free movement. We know that on Sundays he went with a friend to Meiseler, the famous violin virtuoso, with whom he played in a quartet. The fact that the Viennese virtuoso Meiseler accepted János into the quartet testifies to the fact that János was already an excellent violinist at the academy. This is mentioned in all his biographies.
He had no connections with mathematicians, with the exception of Károly Szász, who was teaching in Vienna at the time. But Szász was not a research mathematician. Conversations with him did not contribute to János's scientific development.
Boyai's forced military service had a profound impact on his life because this closed all the doors leading to scientific work. His military service began routinely: after graduating from the military academy with honours, he was appointed a second lieutenant in the Temesvár fortress in 1823. In 1826, he was transferred to Arad, and in 1827, he was promoted to first lieutenant. As a result of his activities in the marshy areas of the Great Plain terrain, he contracted malaria and, as can be assumed from his later complaints, also joint inflammation. The aggravation of these two ailments caused frequent seizures. Taking into account his illness, he was transferred to Lvov in 1830, but he could report to the garrison only in 1831. On the way, he fell ill with cholera, which he mentions in a letter to his brother: "In 1831, on the way to Lvov, I fell ill with cholera in Bistrita ... I felt very weak." [Manuscript Department of the Library of the Hungarian Academy of Sciences. Letter written in 1857.]
János met his father in 1825 during a visit to Marosvásárhely. They met again before János's departure for Lvov, at an unknown location. Both times, they engaged in lively debates over non-Euclidean geometry. Farkas was unable to fully embrace János's overly bold and revolutionary theory, which overturned two thousand years of fossilized scientific prejudice. At their second meeting, they parted on the understanding that Janos would write his theory as an appendix to the Tentamen and that Farkas would send one copy to Gauss: this would let him decide whether Janos's theory was correct or not.
The reason why even such an excellent mathematician as Farkas could not follow János's thoughts is essentially a millennia-old prejudice. This prejudice lies in the metaphysical concept that the world is subject to eternal and immutable laws, and that Euclidean geometry should be the only mathematical reflection of these laws. True, the fundamental tenet of Euclidean geometry, the so-called Fifth Postulate, was criticised even in ancient times, since it is not as obvious as the other axioms. But the result of this criticism was the erroneous conclusion that the content of the Fifth Postulate is not an axiom, but a theorem that can be proven with certainty on the basis of the other axioms of geometry. Consequently, the criticism of the Fifth Postulate did not change the rigid metaphysical conception of the world, according to which Euclidean geometry is necessarily true; its goal was only to find an exact proof to confirm its conviction, considered true a priori.
The geometric content of the Fifth Postulate can be expressed in the simplest way in Proclus's formulation: through a given point in a plane , only one line can be drawn parallel to a given line . This axiom is not, in fact, as obvious as the others, because it is not finite in nature; that is, experiments conducted in a finite part of space cannot be used to verify the axiom's assertion in physical space. This circumstance was the reason that a proof of the Fifth Postulate was sought for over two thousand years, without which Euclidean geometry, and with it the entire worldview of that time, remained without logical justification.
Criticism of the problem of the Fifth Postulate was continued in ancient times by mathematicians of Arabic culture. In Europe, in 1663, Wallis made a great advance in this area by pointing out that the Fifth Postulate is equivalent to the following statement: there exist similar triangles of arbitrary size. His work was continued in 1733 by Saccheri, then Lambert, d'Alembert, Fourier, Monge, Carnot, Laplace, Lagrange, Farkas Boyai, Schweickart, Taurinus, and Gauss, who developed research methods with the goal of resolving this two-thousand-year-old problem. Characteristic of the strength of prejudice is that Lambert, finding no contradiction in assuming the Fifth Postulate was false, did not publish his work, which was published only after his death, in 1786. Taurinus, however, in 1826, could base his conviction that Euclidian geometry was the only correct theory of space solely on philosophical prejudices. But the most characteristic behaviour is that of Gauss, who essentially saw that non-Euclidean geometry, which replaces the Fifth Postulate with another axiom, is logically as equal justified as Euclidean geometry. However, since this idea fundamentally contradicted public opinion, he did not dare publish his results in this area.
The significance of Boyai's work lies in the consistent development of the revolutionary bold idea that the Fifth Postulate can be replaced by the following axiom: through a given point in a plane , one can draw an infinite number of non-intersecting straight lines (parallels) with a straight line , if does not contain . Boyai, in his brilliant work known as the Appendix, drew conclusions from this axiom and showed that Euclidean geometry, based on the Fifth Postulate is logically no more necessary than non-Euclidean geometry. This means that János Boyai, with his correct method, overthrew the two-thousand-year-old prejudices that considered the Euclidean concept to be the only possible a priori, and thus created a new geometric picture of the world, as he wrote to his father on 3 November 1823: "I have created a different world out of nothing." The boldness of his discovery is truly matched by Stalin's characterisation of true science, according to which progressive science: "... must have the courage and determination to break old traditions, norms, and attitudes when they become obsolete, when they become a brake on progress, and can create new traditions, new norms, new attitudes."
Farkas Bolyai, like most mathematicians of that time, could not follow his son's extremely bold theory. His resistance to this discovery is understandable, because the ideas of the Appendix demanded from readers of that time not only mathematical knowledge, but also the acceptance of ideas that were directly opposed to a worldview which had been considered sacred and inviolable for thousands of years. Anyone who wanted to follow non-Euclidean geometry of János Boyai had to accept that the sum of the angles of a triangle is not 180°, but less than that, that only congruent triangles can be similar, that there are no triangles with arbitrarily large areas, that the task of squaring the circle can easily be solved, and that many other astonishing facts are also true. This is precisely why Bolyai's theory required not only mathematical knowledge but also moral strength, capable of casting aside the ideological prejudices of contemporary society.
Individual prints of the Appendix were ready as early as June 1831. It is known that Farkas sent a copy to Gauss on 20 June 1831, but this copy was lost. Gauss received a second copy in January 1832. His reply was written on 6 March 1832. Gauss's reply, as is well known, is very strange. Here is the part pertaining to this matter:
Now a little about your son's work. If I begin by saying that I shouldn't praise him, you'll probably be surprised, but I can't do otherwise, since praising him would mean praising myself, because the entire content of the work, the path of research and the results achieved, almost completely coincide with my reasoning for the last 30-35 years. Indeed, this greatly surprised me. I had intended not to publish this work in my lifetime, about which I have written very little. Most people are incapable of perceiving the essence of this matter, and I have met very few people who would show particular interest in what I told them. This is facilitated only by a very vivid sense of what is actually missing, and this is unclear to most people. However, I had the intention, in time, to write everything so that all my thoughts would not disappear after my death. Therefore, I was very surprised that I no longer have to worry about this and I am very glad that it was the son of my old friend who preceded me in such a wonderful way.This letter, in which Gauss seemingly appropriates the work of János Bolyai, sounds quite astonishing! It is true that, based on his letters, notes, and research in surface theory, we know that Gauss had glimpses of non-Euclidean geometry. But he did not develop his results into a unified theory. Thus, he had no moral basis for writing such a letter, in which he cautiously expresses his recognition, but at the same time claims, to a certain extent, priority. He had no moral basis, if only because it is unknown how far he had advanced in this direction, but the reasons why he did not publish his results in the field of non-Euclidean geometry are known: Gauss, in his own words, feared "the outcry of the Boeotians"; he feared the possible indignation of public opinion caused by the extraordinary boldness of his theory. He knew that such a theory, according to which it is possible that the mathematical representation of the world we have created is incorrect, is revolutionary, at least in the field of science. But the ruling class of the time, during the reign of the Holy Alliance, which was fraught with repression by terror, feared revolution even in the field of science. So, for social reasons, Gauss did not dare publish his results. Let's be honest: the court councillor Gauss lacked the courage to follow the thoughts of the mathematician Gauss.
János Bolyai rightly viewed the insult Gauss inflicted on him primarily from the perspective of the general interest. He rightly wrote that Gauss's cited reasons for refraining from publishing his results "are powerless and ineffective, since in science... it is always a question of clarifying necessary and generally useful, but unclear, matters..." He was absolutely right that the lack of understanding of the most important ideas by people of mediocre abilities "cannot serve as a reason for judicious people to produce superficial and mediocre work and to abandon science in its lethargic, hereditary state..., since life, work, and merit do not consist in this."
However, János Bolyai viewed Gauss's letter not only from the perspective of general scientific interest. By refusing public recognition of his brilliant work, Gauss dealt a severe blow to a man with tense nerves, an ill man who had long desired to be exempt from military service. It would be a mistake to think that Bolyai was only upset by wounded vanity. It was much more than that. He had long hated military service and was reluctant to fulfil his duties even in a good state of health. But he could only be released from military service by being appointed a mathematics teacher instead. Only Gauss's offer of recognition could have ensured this for him. Gauss's letter meant that even after the publication of his brilliant discovery, the path to scientific creativity was not open to Bolyai. Based on the data from military papers, we understand how his disappointment unfolded.
In the garrison where he had previously served, Bolyai apparently had disagreements with some members of the officer corps and, as a result, gained a reputation among them for being quarrelsome. Lieutenant Colonel Zimmer, commander of the Lvov garrison, who was assigned to investigate this matter, characterised János in 1831, i.e., before the arrival of Gauss's reply: " Nothing of this quarrelsome tendency or intolerability can be detected; on the contrary, he is known as a rather shy and very good-natured man." Thus, in 1831, when János Bolyai could hope that, thanks to the success of his work, he would be freed from the drudgery of military service and that, as a mathematics teacher, he would be able to engage in scientific research, he behaved as a " shy and very good-natured man." But when he received Gauss's letter and learned that even his brilliant creativity would not save him from this situation, he became disheartened. The change in his behaviour is clearly shown in his report cards for 1831 and 1832:
| in 1831 | in 1832 |
| Temperament: | |
| calm, good-natured | very irritable, hot-tempered |
| Attitude towards citizens: | |
| good | unknown |
| Behaviour with superiors in military unit: | |
| respectful attitude | avoids all communication with officers |
| Attitude towards subordinates: | |
| good | silent |
| What passions he has: | |
| none | passion for playing chess and devotes a lot of time to this game |
János Bolyai's despair is understandable not only from a human perspective, but even more so because he clearly saw that the significance of non-Euclidean geometry goes beyond the "internal affairs" of mathematics. This discovery fundamentally changed the scientific concept of geometry, but also of the material world in general. Absolute geometry implicitly contains the crucial recognition that any axiom system of geometry does not directly describe the spatial relations of material reality, but is an abstract development of the spatial perspective, which, as a special case, includes the description of real physical space, but is itself capable of further development along with the development of our mathematical and physical knowledge. The replacement of the rigid metaphysical worldview reflected in the belief in the sole correctness of Euclidean geometry with the dialectical worldview of absolute geometry, which encompasses multiple possibilities into a single unity, opened a new scientific era in which even the seemingly definitive axiomatics of geometry are constantly evolving knowledge.
János Bolyai, as we mentioned above, was fully aware of the enormous scientific significance of his discovery, but he also saw its implications for worldview. In the introduction to his 1834 work, The Science of Space, he notes that geometric knowledge does not mean a once-and-for-all, definitive reflection of the material world, but rather, based on knowledge of the nature of a given era, "one must be satisfied with the best possible description." Thus, Bolyai clearly saw that the picture of the world reflected in geometry is not metaphysical, but evolving, dialectical.
One cannot help but dwell, at least for a moment, on the dialectical development of spatial understanding that began with the discoveries of Bolyai and Lobachevsky.
The research of Riemann, Helmholtz, and Lie completed the process begun by Bolyai and Lobachevsky. Without the previous discoveries in non-Euclidean geometry, they would never have achieved these results. Modern differential geometry therefore has a close, almost direct, connection with the ideas of Bolyai and Lobachevsky regarding the foundation of non-Euclidean geometry. However, the mathematical concept of space could only rise to the generalised level of our time when, by the turn of the century, the realisation had matured that the seemingly distinct classical geometry and the theory of point sets were united in the concept of abstract mathematical space. The idea of abstract space in general first appeared in the works of Fréchet, who based the theory of mathematical space partly on the convergence of sequences of points, partly on an axiomatic interpretation of the concept of distance. However, the most general abstract concept of space could only be created after the recognition that the only characteristic of a topological space is that it can distinguish points of concentrations and isolated points. This decisive beginning was laid by Friedes Riesz. The subsequent development is very branched: however, an important result of his work is the creation of a connection between different types of abstract spaces, which is expressed most clearly in the famous Urysohn theorem. According to this theorem (in an extended form by Tikhonov), every regular topological space with a countable basis is homeomorphic to the basic rectangle of a Hilbert space. Thus, Urysohn's theorem eliminated the sharp boundary between abstract topological space and the more visual coordinate space.
János Bolyai's thoughts on the structure of space contributed to the extraordinary development of spatial science. This development, on the one hand, led to even seemingly amorphous sets being subsumed under the concept of mathematical space; on the other, it enabled the characterisation of Euclidean and non-Euclidean spaces with more intuitive geometric properties within a broad class of metric spaces. To mention one of these results, the following concept is necessary: we call a point the centre of a pair of points if the distances between them satisfy the condition \overline{pr} = \overline{qr} = \large\frac{1,2}\normalsize \overline{pq}. We will call the metric space regularly convex if any pair of points has one and only one centre, and for each pair of points there exists a point such that the point is the centre of the points. Busemann showed that three-dimensional non-Euclidean spaces with the Minkowski metric coincide with those three-dimensional regularly convex complete spaces that, in addition, satisfy the conditions that Busemann called the axioms of the boundary circle.
The foundation of this enormous development of the concept of space is Bolyai's world of ideas. This rich treasury of new mathematical concepts also provided us with the tools necessary for creating a modern physical understanding of the world. Thanks to this, it became possible to mathematically present numerous sections of the general theory of relativity and quantum mechanics.
In 1832, János Bolyai was promoted to captain of the second rank and transferred to Olmütz. During the journey, his carriage overturned, and the ailing Bolyai suffered a concussion. At the direction of the High Command, he underwent a medical examination, and by order of 28 May 1833, he was discharged with a salary of 280 forints per year, which was to be paid in monthly instalments by the Sibiu war fund. Thus ended János Bolyai's military career, and on 16 June 1833 he returned home to Marosvásárhely.
He stayed with his father in Marosvásárhely for almost a year. During this time, heated arguments broke out between father and son. We know nothing about the nature of these squabbles, but several clues suggest that they arose partly from academic and partly from ideological differences. Farkas was fundamentally a progressive man, influenced by the rational spirit of the Encyclopaedists, but his everyday circumstances forced him not to defend his views. János, on the contrary, expressed his radical opinions with unbridled frankness in all matters, firmly insisting that a person convinced of his rights must boldly defend his ideas against any opponent. It's clear that Farkas, whom the obtuse petty bourgeoisie of Marosvásárhely had incited against his "ungrateful and callous son," couldn't tolerate János's behaviour, which over time could even threaten Farkas's social standing. Following a heated argument, Farkas kicked János out of his home. Some time later, at the request of his brother, Antal Bolyai, he agreed to his resettlement in the village of Domald to manage Farkas's farm estate. This was something János desperately needed, as his pension of 280 forints a year barely covered his basic needs.
In 1834, János Bolyai moved to the village of Domáld, a tiny village in the Kishküküle region of Transylvania, where he lived until 1846, a total of 12 years as a rural hermit, far removed from the cultural world. However, despite the lack of financial support, János Bolyai continued his work in Domáld. It was during this time that he submitted his work, Responsio, to a competition organised in Leipzig by the Jablonowski Society for the development of the theory of complex numbers.
The writing of the Responsio was preceded by a correspondence with Farkas containing a scientific discussion, in which János pointed out to Farkas an error in his theory on the justification of complex numbers. He ended his letter with the following words: "Now you cannot help but be convinced that your theory has been refuted. Clearing up any persistent ambiguities or errors is a pleasure and delight for me, and I suspect it's the same for you." However, "clearing up ambiguities" didn't mean "pleasure and delight" for Farkas, resulting in a special, one might even say sporting, rivalry between father and son regarding the competition. Both submitted their competition entries to Leipzig, but neither received a prize. Half the prize was awarded to Ferenc Kerekes, a professor at the Debrecen College, whose competition entry was formulaic.
Written in 1837, the eight-page Responsio resembles more of a sketch than a mere outline, although it contains many profound ideas. János Bolyai defined complex numbers by introducing four units, pointing out that a similar theory cannot be constructed with more than four units, since in that case, for example, the square root would not be two-valued. Consequently, János Bolyai developed this theory in the same year that Hamilton created the theory of quaternions. Bolyai's profound mathematical insight is demonstrated by the fact that in the Responsio he interpreted the power for a complex exponent with an Maclaurin series, which is already present in Euler, but Bolyai also uses it to interpret the logarithm in the case of a complex argument as its inverse function. There is no doubt that, despite all its crudeness, the Responsio would be enough in itself to secure János Bolyai a place in the history of mathematics.
Farkas was dissatisfied with János's farming activities in the village of Domáld. In 1846, he expelled him from the estate, which he had leased out. János and his family settled in Marosvásárhely, where he built himself a house with the money he had saved in Domáld.
In Marosvásárhely, János Bolyai lived in complete seclusion, almost like Robinson Crusoe on his island. It is characteristic of his isolation from public life that his friend from his youth, Károl Szász, during his four-year stay in Marosvásárhely, "did not visit him, as he did not dare to do so, so as not to compromise himself." The conservative bourgeoisie of Marosvásárhely feared the bold and progressive János Bolyai so much that his famous German biographer, Steckel, who stayed there at the beginning of the century, wrote the following: "Even forty years after his death, one could observe how hated and despised the name of János Bolyai was in Marosvásárhely."
The year 1948 holds a special significance in Boyai's life. The events of the War of Independence brought him out of his isolation. As a former officer, he would have eagerly enlisted in the military, but constant illness prevented him from doing so. In one of his letters from the beginning of 1848, one can read the following: ".. due to my weak physical condition ... I am completely unfit for military life, but I try to advance the common cause at least through personal services ... for as long as there is even one dissatisfied person, no one can boast of a completely happy state." However, when the Székely troops won a victory at Radnót in October and on 30 October several people gathered for a secret meeting in the apartment of Hussar Major Šánta to discuss further military actions, János Bolyai also left his sick bed. In one excerpt from his diary, the Transylvanian historian Farkas Deák relates that at this meeting, "Bolyai presented a clear and concrete plan providing for the cleansing of the region of Szeben, Fehérvár and all of Transylvania..., promising to hand over all of Transylvania to the Hungarian government by the new year." "This plan was not adopted," Deák continues, "because the leaders and commanders, out of pride, rejected János Bolyai, and mainly because Dreschner was a reactionary and Zsombori was vacillating, and therefore the two experts sought to refute this best possible plan." It is striking how accurately Deák describes in his diary a characteristic feature of the history of revolutions: the enemy places his chosen agents in the general staff of the revolution, who, under the guise of specialists, prove that good things are bad, and vice versa, thereby causing catastrophes. The same thing happened in Marosvásárhely: the undisciplined Székely units were routed by the Emperor's regular troops, and the Austrian Lieutenant General Gedeon entered Marosvásárhely, permitting free plunder, and János Bolyai, as Deák writes, "... returned to the solitude he had just left."
János Bolyai, particularly inspired by Böhm's successes, repeatedly wanted to go to the battlefield, but illness always prevented him from doing so. During the repression after the War of Independence, he completely withdrew from society. But even during the War of Independence, he came across a scientific work that piqued his interest in an extraordinary way. This was Lobachevsky's Geometrische Untersuchungen zur Theorie der Parallellinien, published in 1840.
Bolyai received Lobachevsky's work from Farkas, who discovered it by chance. After reading it, he was surprised to discover that its contents were similar to Appendix. At first, he suspected that Lobachevsky didn't exist, that Gauss was hiding under this pseudonym, and that the work was a plagiarism of the Appendix. This conclusion was supported by Gauss's alleged behaviour, in which he supposedly praised Lobachevsky's work while remaining silent about the Appendix. Whether this was true or not - which is more likely - János was led to believe that this is precisely what happened. Under these circumstances, he certainly had every right to suspect that the Geometrische Untersuchungen had been created as a result of Gauss's appropriation of his ideas. This suspicion is not difficult to understand if we consider that János Bolyai had lived alone for several decades, far from the scientific world, unaware of the developments in the problem prior to the publication of the Appendix. And suddenly he learned that, after two thousand years of failure, a third creator of non-Euclidean geometry had appeared. Faced with this incomprehensible fact, and outraged by Gauss's earlier real and later perceived behaviour, it is entirely understandable that he first thought of plagiarism. That is why he made careful notes on Lobachevsky's work to discover whether there were any errors in it that would reveal a possible plagiarism. Bolyai, although he harshly criticised the minor and major errors in Lobachevsky's work in his notes, himself recognised that Lobachevsky was a genius like him. In one of his observations, for example, he expressed his appreciation for Lobachevsky with the following, in the style of a sly judge:
Walking on such edges and standing on such heights, Lobachevsky very beautifully and nobly, eminently, and with great poise in his main idea, like the greatest, most skilful and most delicate artists dancing on ropes and wires, expounds the self-reliance of the theory of spherical triangles.As for the priority of these two brilliant authors, the only opinion that can be drawn is that the question of priority is inappropriate to even raise here. Undoubtedly, Lobachevsky's work was published two years earlier than Bolyai's. However, we know that Bolyai discovered the most essential fundamental idea as early as 1823, when he wrote to his father that he had "created a new world out of nothing," while Lobachevsky had not yet found his way to the solution, as evidenced by his manuscripts of 1823. But in 1825, both wrote down their results: Bolyai submitted his manuscript in early 1826 to his mathematics teacher, Captain Wolter, and Lobachevsky handed over his manuscript to the Physics and Mathematics Department of Kazan University in early 1826. Both manuscripts were lost. This brilliantly demonstrates that both geniuses, essentially at the same time, independently of each other, discovered non-Euclidean geometry.
Their method of interpreting their work naturally differs. Bolyai treats several construction problems, proving that in non-Euclidean geometry, squaring the circle is solvable, while Lobachevsky sought first and foremost, to find formulas for resolving ordinary geometric calculations in non-Euclidean geometry. These differences in interpretation can be explained by the social status of the two scholars. Bolyai, a reclusive man, had only paper and ink at his disposal, so he focused on theoretical conclusions that he could verify even in his seclusion. But Lobachevsky, as a university professor and an educated astronomer, hoped from the outset to determine empirically the Euclidean or non-Euclidean nature of physical space, and therefore primarily sought tools that astronomical practice could use to address this issue.
Lobachevsky's work sparked Bolyai's interest in geometry: in 1850, he began working on the manuscript The Theory of Space, to which he returned several times until 1855. From the fragment that has survived, as well as from notes whose contents are known to us, it is clear that János Bolyai explored significant new ideas in this work as well. The most significant part of the fragment is that in which he deals with the general theory of surfaces and curves. This demonstrates that Bolyai, ahead of many other researchers, recognised the importance of the topology of surfaces and curves. He divided curves into simple and knotted. He set himself the goal of compiling a list of types of figures, which he called simple surfaces. He apparently used the term "simple surface" to refer to two-dimensional manifolds. Bolyai distinguished between complete surfaces and surfaces with a hole: this indicates his desire to characterise surface types by their connectivity. His next remark demonstrates a very fine geometric sense: "On any simple surface, one can make an arbitrary number of indentations, place tubes in these places, and connect them in pairs. A simple surface has this property in the most general case." And he adds: "The proof of this must be examined!"
Bolyai's remark regarding the proof of Euler's theorem on polyhedra is absolutely correct, according to which "the theorem of the great Euler... is not proved in a sufficiently general form, because not every relation of polyhedra is obtained by cutting off pyramids." Unfortunately, Bolyai's proof is unknown to us, and therefore we do not know for which multiply connected polyhedra he managed to generalise Euler's theorem when he said that he had "found a proof of Euler's relation for the cases of ring-shaped polyhedra and hollow-plane spaces."
In 1856, Bolyai again began to study the question of whether non-Euclidean geometry is consistent in three-dimensional space. He clearly saw that even if flat Euclidean and non-Euclidean geometry are both consistent, one cannot yet draw conclusions about Euclidean or non-Euclidean relations in three-dimensional space. In this regard, Bolyai went somewhat further than Lobachevsky, who was satisfied with the fact that the geometry of the non-Euclidean plane is consistent. Bolyai wanted to solve the problem as follows: if we take a tetrahedron with vertices and take a point outside it, then these five points determine 10 edges, 10 triangles and 30 angles between the faces. By applying absolute trigonometric formulas to the trigonometric relations of the latter, he hoped to resolve the question of consistency in this way. For a time he was mistaken, since, as a result of one error, he thought that he had succeeded in proving, on the basis of constructions of three-dimensional space, that the Fifth Postulate, or otherwise known as the Eleventh Axiom, is necessarily true. He even began writing an essay titled: Proof of the universal significance of the eleventh Euclidean axiom, hitherto unproven and serving as the basis for the entire science of space and motion. Written by János Bolya retired Captain Engineer. But he didn't write beyond an introduction after noticing an error. Until the end of his life, he failed to prove the consistency of non-Euclidean geometry. This, as we know, was proven after his death.
After the death of Farkas Bolyai in 1856, János moved in 1857 to the outskirts of the city, to Kalvária Street, near the Catholic cemetery, where he rented a very small house. Here he lived alone, incapacitated, constantly ill and completely unable to work. Only to his brother Gergely did he occasionally write letters about his family affairs and his illness. In 1857, recalling the pleasures of scientific work, he wrote:
... I came to know a paradise which, despite all the severity of my way of life, I would not have given up for any material thing in the world and which anyone can come to know by working diligently and with inspiration to improve themselves." In later years, he spent almost all his time in bed and had a fever. He was constantly cold, and even in the summer he slept in a sheepskin hat. We do not know what he suffered from. One of his complaints was that both his legs were swollen and painful from the ankles up, perhaps a symptom of advanced arthritis. On 18 January 1860, he fell ill with pneumonia and soon after began to approach death. On 27 January, his maid, Juliana Szöts, who ran the household, wrote the following tragic letter to Gergely Bolyai: "I wanted to inform you earlier about the captain's condition, but when he was feeling better, he did not allow me to write to you. The Captain fell ill on 18 January, but he kept talking. I asked him many times to write but he wouldn't because he was afraid of the expenses. But today, 26 January, at 10 o'clock in the evening, he lost his speech and lost consciousness, I expect him to die any minute. I ask, Your Excellency, to hurry to your brother, God only knows whether you will find him alive, besides, it is unknown with whom he kept his money, because he never told us about it. The maid, Juliana Szöts, respectfully asks you to hurry.Apart from the obligatory military delegation, only three people attended his funeral: his grave remained unmarked. When Ferenc Schmidt, the first Hungarian researcher of the Bolyai family, searched for his unrecognisable grave in 1893, only one person, the old servant Julianna Szöts, could show him János Bolyai's grave.
While I was writing the letter, the captain died, so he is no more.
Bolyai's life is truly a tragedy, because his remarkable discovery remained long unappreciated, and he was forced to ponder, alone, a world in which reason and science would triumph over senseless violence. The crucial reason for his failure is that in the first half of the 19th century, scientific research in Hungary, and even throughout the Habsburg Monarchy, lacked a public foundation. The exact sciences, especially mathematics, a science of an abstract nature, are important only in a society whose mode of production necessitates the direct or indirect application of mathematical advances in production. It is no coincidence that it was the Greeks, a people of industry and a seafaring people, who developed mathematics to such a high level in ancient times. It is no coincidence that in the 17th century, it was in England and France - where industrial development and the exploitation of geographical discoveries were the most important social demands - that modern mathematics began to develop on such a grand scale. This means that mathematics was of interest to those societies whose ideology was determined by the productive forces and advanced production relations developed for a given era.
Characteristic of the backwardness of productive forces during Bolyai's lifetime is the fact that in 1841, Hungary had only six steam engines with a total capacity of 74 horsepower, while Austrian industry had 231 steam engines, and France had about 4,000. Our source of energy was the exploitation of serfs and farm labourers. Under feudal conditions, industry consisted of moribund guilds or, at most, a few manufactories that played a completely subordinate role.
The backwardness of the productive forces and the associated production relations of Hungarian society at that time created an ideology in which the exact sciences and research could barely develop. The ruling class, defending feudal exploitation to the extreme, supported neither exact sciences research nor those who based their worldview on the achievements of exact natural science. And János Bolyai did just that. His desire was not only to engage in mathematical research, but to solve all questions, including social problems, using scientific methods. As a result of this behaviour, sooner or later he was forced to clash with everyone who subscribed to the then-accepted worldview. Consequently, the tragic course of János Bolyai's life is not the result of personal relationships and individual qualities, but the suffering of a great creative genius and supporter of radical reforms, struggling in the grip of the backward Hungarian social system of that time.
János Bolyai was a thinker interested in philosophy. In addition to mathematics, he explored natural philosophy and social issues. His bold thinking reveals him as a consistent rationalist who, particularly in the field of mathematical philosophy, advanced essentially materialistic ideas.
It deserves special attention, for example, that János Bolyai, speaking about the theory of complex numbers in §11 of his Responsio, writes the following:
... only such things, and thus only such quantities, can be the objects of sound research that really exists (e.g. material, part of the physical or external world, or at least conceivable and possible).Thus, Bolyai opposed the arbitrary creation of mathematical concepts, and in regard to mathematical idealism or materialism, he decisively sided with materialism. He considered complex numbers to be a unique reflection of reality, despite the prevailing belief in his time that complex numbers were imaginary objects that did not describe material reality.
However, not only some individual notes by János Bolyai, but also the entire content of the Appendix, his view of mathematical space, testify to the fact that his scientific method is essentially materialistic. Even about the origin of his geometry, he writes that he had to decide on the rejection of the Euclidean view "I believe that nature cannot be forced, nature cannot be formed according to fancies born of dreams, and the truth, or nature itself, must be revealed in a rational and natural way, and it is necessary to be satisfied with the best possible examination." Consequently, Bolyai considered the world to be an objective reality, the description of which is the task of mathematics. This description should not express truths formulated arbitrarily "according to fancies born of dreams," but must express objective truths independent of our consciousness. This point of view corresponds to the view considered by Engels as the criterion of materialism: "the materialistic worldview, says Engels, means simply understanding nature as it is, without any extraneous additions." It should be mentioned that Bolyai knew that our concepts, which serve as a reflection of nature, also develop, which means that in any specific historical situation "it is necessary to be satisfied with the best possible consideration."
The consistency of non-Euclidean space dealt a severe blow to Kant's idealistic theory, which considered our concept of space a priori, and therefore its mathematical picture final and incapable of further development. However, Bolyai's discovery, which resulted in the expansion of the mathematical concept of space, confirmed that both the Euclidean and non-Euclidean character of physical space are equally possible a priori, and therefore Kant's idealism, or any other metaphysical conception of physical space, is incorrect. This means that the mathematical content of the Appendix alone is a significant step toward dialectical materialism.
Bolyai consciously waged a struggle against the emptiness of metaphysical phrases. He writes in his work Raumlehre the following: "Scholastic metaphysics is for the most part the miserable offspring of overstrained, diseased forces, insufficiently versed in the field of human knowledge, since what is precisely established by science pertains to mathematics as the only truly fundamental science; everything else is pure pettiness, distracting us from the majestic fields of the most useful and most fertile sciences." This remark is a criticism of metaphysics, although not a materialistic criticism. Boyai, as a rationalist, expected everything from mathematics.
It is characteristic of Boyai's worldview that, like Laplace, he considered the world to be a single whole, in which each part is in a lawful relationship with every other part of the world, so that their movements mutually determine each other. "Between the parts of the entire world", writes Bolyai, "there is a necessary and strict regularity. Without any particle, even without a speck of dust, the world could not exist ... on the other hand, any large part, even the entire world without a speck of dust, is insufficient to regulate the course of the entire world." It is interesting to compare these thoughts of Bolyai with the teachings of dialectical materialism, according to which dialectic views nature not as a random accumulation of objects and phenomena separated from each other, isolated from each other - but as a coherent whole, where objects and phenomena are organically connected to each other, depend on each other and condition each other.
The rationalistic basis of Bolyai's ideas, of course, weakens the consistency of his materialistic views. This is evident from his notes on the question of the objective existence of space and time. He writes: "Whether space and time actually exist or only in our consciousness should be a matter of indifference to an intelligent and judicious philosopher, and should remain outside of consideration to the same extent that this question, in the strictest sense, cannot be resolved; however, these ideas seem to be such that they must relate to actually existing things, since apparently the idea of the existence of space and time is correct and to doubt it would mean going beyond all limits ..." Thus, consistent rationalism forces Bolyai to hesitate in the question of the objective existence of time and space, since he wants to resolve this question rationalistically, while this is "in the strictest sense," i.e., mathematically impossible. Indeed, the formally mathematical method will lead to the same results, whether we consider space real or apparent. But Bolyai did not get stuck in rationalistic doubts, but took a step forward to a materialistic view, since, in his opinion, to doubt the objective existence of time and space would mean "going beyond all limits," that is, our worldview would become irrational if we "raised further doubts" regarding the objective existence of space and time.
Bolyai was also interested in social issues, especially toward the end of his life (1850-1855). He drafted a plan for reforms that seemed too extreme, even insane, to his contemporaries. He considered composing a "Doctrine of the Good," which would contain, on the one hand, a utopian-socialist reform of society, and, on the other, an encyclopaedia of sciences reminiscent of Comte's ideas. He did not even complete the outline of this work; we can reconstruct its contents only from his notes and the opinions of his contemporaries.
The progressive nature of Bolyai's plans for social reform lies in the fact that he recognised the misery arising from class oppression, and he did not intend, as was the custom of that time, to get away with pious phrases and promises at the promise of the afterlife, but sought a method for overcoming poverty as soon as possible. In his opinion, "society is full of misery and misfortune, but it does not have to be so. One can be happy on earth." According to him, the main reason for the misfortune of humanity is the selfish individualism of a few people, which must be replaced by the observance of common interests. In one note he says: "... it is sad and doubtful... but also dangerous for oneself ... to want to build one's personal happiness on the basis of the oppression of others." Elsewhere he writes: "There is no personal happiness without the happiness of the whole society."
János Bolyai believed that production should be carried out jointly, based on the division of labour and shared ownership, and that distribution of goods should be based on the principles of general equality. He believed that such a socialist society would be created through moral influence and persuasion. " ... Only spiritual force, i.e. persuasion, victory and voluntary transformation, or birth, can lead to salvation and we hope that at least before the year 2000 - without any dreams, of course - this hour will come," he wrote in one of his letters. The knowledge and mastering of the "Teaching of Salvation" by people and the subsequent improvement of their morals would lead to "voluntary transformation". "A significant and essential part of the salvation is the mastering of the teaching of salvation, another, no less noble and distinguished, is having a way of life that corresponds to the doctrine of salvation"
According to these views, János Bolyai was a utopian socialist with all the virtues and shortcomings of utopians. The virtue of his ideas on transforming social order lies in the fact that he wants to put an end to exploitation and class oppression. This is precisely why his reformist ideas are progressive. His weakness - naivety - is the result of ignorance of the material forces that determine the movement of society. The reason for this naivety is the same as for all other utopian dreams: "This fantastic description of a future society arises at a time when the proletariat is still in a very undeveloped state and therefore still imagines its own situation in a fantastic way; it arises from the first premonition of the proletariat's impulse toward the general transformation of society" (Marx-Engels).
One hundred and fifty years have passed since the birth of János Bolyai. During this time, in Hungary and Romania, where he was born, a world has been realised under the leadership of the proletariat in which it is impossible to "build one's personal happiness on the basis of oppression." In this new world, the role of science and the scientist has changed radically. Gone is the era in which, using Bolyai's own words, "the working people were dominated by people who worshiped the golden idol and the money-god, believing and trusting only in money and listening only to its music." Where nations have taken power into their own hands, science is the common cause of the entire people. There, scientists create not for a few, but for the entire people. Bolyai today already belongs to all progressive humanity. In the Soviet Union, the Appendix was published; in our country of Hungary, it was recently released. In the Romanian People's Republic, a university is named after János Bolyai, and here in Szeged, we have a János Bolyai Mathematical Institute and Mathematical Society. In Romania, we are solemnly celebrating the 150th anniversary of his birth. Our working class is familiar with his achievements through lectures given in factories and plants, and millions of workers respect him. Before us, Bolyai appears as he truly was: a true scholar who fought for the cause of truth. He was a giant among scholars. He always sought the truth, which ultimately serves the working people. And the liberated nation, 150 years later, remembers with fervent affection its long-suffering, brilliant son who suffered so much for the truth.
Last Updated March 2026