**János Bolyai (1802-1860)**

## Imperial-Royal captain of engineer corps and the Mozart of mathematics (geometry)

### by Harald Pöcher

*“Semmiböl egy újj más világot teremtettem!”*

*“I have created a new world out of nothing!”*

** **

János Bolyai, the Hungarian mathematician and captain of the Imperia-Royal military engineer corps was one of the founding fathers of Non-Euclidian geometry. With his ground breaking discovery of the Non-Euclidian geometry he laid the basis for other scientific disciplines, i.e. the theory of relativity, which was developed by Albert Einstein in the 20 Century. To lifetime he had no appreciation, but nowadays he is considered the most important mathematician (geometrician) of history of mankind.

Because he was educated at the Imperial-Royal military engineer academy in Vienna, which was located in the nowadays Stiftskaserne (Stifts-Barracks), the memory at him was held up by the Hungarian Forces (Honvéd), the Faculty of Military Science and Officers Training at University of Public Administration in Budapest (Nemzeti Közsolgálati egyetemi hadtudományi és honvédtisztépzö kar), the János Bolyia society (Bolyai János alapitvány) and the Austrian armed forces, especially the National Defence Academy. The annual ceremony in the presence of high ranking representatives of Austria and Hungary in the Stiftskirche (Stifts-Church) and Stiftskaserne (Stifts-Barracks) to his memory is just one of many events to duly honor János Bolyai. The essay gives an insight into the life and work of János Bolyai.

**Childhood and youth years**

János Bolyai was born on 15 December 1802 in Klausenburg, the former Kolozsvár and present-day Romanian Cluj-Napoca in a Calvinist household as son of Farkas Bolyai (1775-1856) and Zsuzsanna Benkö de Arkás (1778 or 1780-1821). His father was a professor of mathematics and a friend of the German mathematician Carl Friedrich Gauß (1777-1855). He worked on the Euclidian geometry. His mother is treated as a beautiful appearance. However, she has suffered since her early youth to hysteria. The father tried hard to the upbringing of his son, and hired the best tutors who should teach János the most important subjects except mathematics, which the father taught himself.

At the age of twelve, János Bolyai passed a rigorosum about the school curricula of six classes. After that he became a “student” in accordance with the customs of that time. “Student”, he has stayed for two years, but he rarely attended the lectures and dealt better with the board game “Dame”. From his truancy some anecdotes remained. His teacher, the bishop János Antal complained to the father, that the boy skipped lectures. The father brought the son to justice but that hardly bare fruits. János did not show discernment after the scolding of the father. Before a test he read the textbooks a few times and passed the exam excellently. Especially, in mathematics János Bolyai was an excellent student who was more than worthily able to represent his father in the class when his father was ill. It is reported that the students rather preferred the lecture of the thirteen years old boy than from the father, because János was able to explain mathematics better than the father. Since his early childhood he loves to play the violin. At the age of twelve he was able to play the most difficult musical works *“prima vista (=without having seen it previously)”.*

The desire of the father was that his son should continue his studies in Göttingen with Gauss. Farkas therefore wrote a letter to Gauß in 1816 asking him, if he would take his son to him within the next three years to teach him mathematics. But for understandable reason he got no answer. After this failure and not least for reason of costs, Farkas saw the only way out to send the son to the Imperial-Royal military engineer academy in Vienna.

**János Bolyai as a cadet at the Imperia-Royal military engineer academy**

In August 1818, János Bolyai entered the academy, which had a high national and international reputation. As is apparent from the documents, the talent of János Bolyai for mathematics was already noticed in the third class because his performance in the subject *arithmetic *and *algebra* was valued with “first class with merit” by the professors. In the fourth class he was trained in *German, French, nice writing, religion, arithmetic* and *algebra*, *simple geometry, history, geography and drawing figures*. In most of these subjects János was valued with “first class”, respective “first class with merit”. Just in *nice writing* he was valued with “good” and in *drawing figures* “slowly”.

In a story it is handed-down what a genius János Bolyai was. During the school year the then head of both military academies and General Director of engineers Arc duke Johann (1782-1859) visited the Imperial-Royal military engineer academy and let call one of the new students. The choice fell on János Bolyai who solved the provided task in record time and immediately turned to solve a new task. The arc duke was amazed at such a genius that he said to the professor: *“The other students should be subordinated to János Bolyai because he knows more than the whole class”.* János Bolyai finished the fourth class as one of the three best. In the fifth class he had to learn the new subjects *higher geometry* and *mathematic geography*. In all subjects except *drawing figures* and *drawing a situation* he was one of the best. In the sixth class he had to learn *mechanics, experimental physics *and* perspective*. In the sixth class he improved his skills in *drawing situations* and got therefore “first class”. In the seventh class in the school year 1821/1822 he had to learn *geometrical drawing*, *architecture in theory and drawing*, *tactics* and *fortification* which he throughout finished with “first class”. From the eight class, the last school year, a notice preserved in which we can learn something about the unfortunately temperament of János Bolyai. According to the order from 23 July 1823, János Bolyai was penalised with house arrest, because he overstayed his leave more times. More sharply formulated was the order from 10 August *“because of his stubbornness, Cadet Bolyai who makes a game out of it to disregard commands and orders, especially the non-observance of the tap, will be punished with house arrest in the amount of the whole month of August”.*

Despite these failings, János Bolyai was promoted to Second Lieutenant on a proposal of Lieutenant field marshal August Freyherr von Herzogenberg, the commander of the academy, by decree of Arc duke Johann of 1. September 1823 *“in consideration of his abilities and the acquired knowledge and of proven good morale”, *and was transferred to the fortification direction in Temesvár. Afterwards his real military careers began.

**János Bolyai as an officer of the Imperial-Royal engineer corps**

János Bolyai began the then difficult journey to Temesvár on 17 September 1823 and arrived at the fortification direction on 30 September. The journey took him through the following cities: 18 September (Bruck an der Leitha), 19 (Györ), 20 (Neszély), 21 (Dorog), 22 (Pest), 23-25 (Stay in Pest), 26 (Orkény), 27 (Filegháza), 28 (Szeged), 29 (Szent Miklós), 30 (Temesvár). After almost three years of service in Temesvár he was transferred to Arad where he stayed at the fortification direction until 8 December 1830. On 8 September 1827 he was promoted to First Lieutenant.

In the confidential reports, János Bolyai was characterized as an ascetic, who drinks no alcohol, has no passion for gambling and who does not seek dispute. However, he was once irritated, he was forced to challenge his opponent to a duel. János Bolyai was very skilful with saber and epee. It was reported that János defeated in only one day thirteen cavalry officers during duels. He set only the condition that it is allowed to him to play his beloved violin for recreation after the completion of two duels. His military abilities were characterized rather average, which is reported in the assessment as *“little usable in service and in practical terms not an officer of engineer corps”.*

At the end of 1830, János Bolyai was transferred to Lemberg, where he arrived six month later due to illness. During the six months of illness he lived at his father’s house in Marosvásarhely and wrote his famous 28 pages appendix *“Scientia spatii absolute vera/The absolutely true science of space”. *The appendix was published in his father’s work “Tantamen” in 1831. From his time in Lemberg it was reported, that due to his often illness, János was most qualified in German, Hungarian, Latin and Mathematics, but he is more professionally qualified for a professor than for the military duty.

As the father read the scientific essay of his son he understood the importance of the scientific discovery. As a result of the work of János, Farkas wrote a letter to Gauß telling him the discovery of the Non-Euclidian geometry by his son. The answer of Gauß still arrived in 1831 and it was clearly and categorically: *“If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my mediations, which have occupied my mind partly for the last thirty or thirty-five years. My interest actually was not to publish the results of my research work during my lifetime. But it was my intention to write everything down so it does not under go with me. I am therefore very surprised that my efforts have been spared me and I am delighted that it is precisely the son of my old friend who is pre-empted me so strangely.”*

* *

The concurring statements of Gauß although raised the self-confidence of János Bolyai, but otherwise János understood that Gauß discovered himself the new geometric relationship and Gauß actually have the right to the first discovery. This fact had far-reaching consequences on the character of János in the following years. He became more and more irritable and choleric irascible. Nevertheless, János Bolyai didn’t loose the benevolence of his commanders because his discovery has even reached the Arch duke Johann. After a service period of one and a half year János was transferred to the fortification direction of Olomouc. And, in Olomouc the military assessment of János Bolyai was not flattering, too. In the assessment the commanders emphasized, that János shows no significant solicitude. Nevertheless, due to his scientific discovery, he is considered for a promotion if he will increase eagerness. In 1832, he was promoted to captain.

Not later than 1832, János Bolyai was tired of the service and he wrote a long letter to the Arch duke Johann which contained the German version of the first 33 paragraphs of his appendix and an abstract of the letter of Gauß. In the letter János Bolyai petitioned the arch duke to free him three years of service so that he could devote himself to reformation of mathematics. The arch duke wrote on the letter in his handwriting: *“Agreed! But it would be best for the service if he could get a job where he is able to use all his skills for teaching mathematics.”* What the archduke meant was that János Bolyai should be transferred to the Imperial-Royal military engineer academy as a professor. However, it remained in the written notice and the arch duke set no further steps to transfer János Bolyai to the academy.

The further military career of János Bolyai decided to the coincidence of several incidents. 1832, János Bolyai became more and more a hypochondriac. Due to a doctor’s attestation, János Bolyai suffered from high-grade hypochondria, defective sight, digestive problems, emaciation and hyperhidrosis. Due to the medical attestation of a staff surgeon, János Bolyai was assessed as unfit for the military service and on 16 June 1833 he was dismissed as half-invalid by pointing out that if his health situation should improve János Bolyai could be employed again as a civilian employee. This incident ended the military service of János Bolyai like so many other hopefully careers of officers of the Imperial-Royal armed forces of Habsburg Monarchy. With some little good will of the commanders, János Bolyai could have been transferred to the Imperial-Royal military engineer academy, where he could have been worked for many years successfully as a professor of mathematics. János Bolyai as a professor for mathematics at Imperial-Royal military engineer academy would have been able to teach generations of engineer officers and he thereby would improve the quality of mathematics in the Austrian-Hungarian Army.

**The life and creative work of János Bolyai as a retired captain of engineer corps**

After his retirement, János Bolyai lived with his father in Marosvásarhely, which is present day Târgu Mureș** **in Romania. However, both often quarrelled and therefore János very soon moved to the family owned manor in Domóld, where he lived more than 10 years in utter solitude in a life partnership with Rozália Orbán de Kibed and the common daughters Amália, Klára-Eliza and the son Dénes under ordinary circumstances. János was not allowed to marry Rozália because János didn’t get the agreement of the military authority due to lack of personnel wealth and money. In 1843 János returned to his father’s house, but very soon afterwards they quarrelled again and János returned to Domóld. And finally, in 1846 János returned to Marosvásarhely in his own house, but his relationship with his father always remained tensely.

During the Hungarian fight for freedom 1848/49 János Bolyai was asked by the Honvéd to join the armed forces. He refused with reference to his bad health but agreed to join the movement as a teacher. János never overcome the defeat of Hungarian forces in the fight for freedom, but in 1849, after Hungary declared its independence, János was able to marry Rozália. The Austrian military authority didn’t initiate proceedings against János, but, if his writings about the fight for freedom would become known by the authority, the Austrian military authority had to initiate a process against János Bolyai.

At the end of August 1849, János Bolyai obtained at k.k. General command for Transylvania in Nagyszeben (Hermannstadt/Sibiu) the payment of his pension. His efforts for recognition of his marriage remained unsuccessfully. In 1852, he separated from his wife, but he took care of his children heartwarmingly. János Bolyai died on 27 January 1860. His grave is located at the protestant cemetery of Marosvásarhely. The grave remained 30 years forgotten, and not later than in 1894 the Hungarian society for mathematics and physics put down a grave stone. His last years as a retired captain of engineer corps was intended by addressing the theory of space and the development of an independent philosophy. Despite the discovery he was not rich and always suffered short of money which moved him to go to the military authority requesting for assistance.

**The “true face” of János Bolyai**

We know little of the outfit of János Bolyai. He was of medium height, and despite of his rural clothing he had a military correct posture. A great mystery surrounds the appearance of his face since no portrait has remained received from him. Often copied images, i.e. an Hungarian post-stamp which was published on the occasion of the 100 anniversary of his death, doesn’t show the true face of the great mathematician. Solely, the bust in Târgu Mureș** **might describe the facial features of János Bolyai, because the bust was purpose-built in 1911 according to the specification of his son. Many artists tried to reconstruct the face of János Bolyai. Such an attempt was made by the Hungarian painter Attila Zsigmond in the 1990ies.

**Euclidian versus Non-Euclidian Geometry**

A presentation of the life and work of János Bolyai would be incomplete, if the worth of his discovery wouldn’t be discussed. Mathematics as a science discipline in the modern sense began in the fifth century before Christ when it was taken over from the Egyptian and Babylonian culture into the culture of Greece. A special feature of this was that during these times the geometry had a higher importance than the arithmetic which then occurred in the guise of geometrics. The history of Non-Euclidian geometry thus began about 2,300 years ago when the Greek mathematician Euclid in Alexandria founded around 325 B.C. with the results of his research work the basis of geometry. In his work *“Stoichea (=elements)”,* he summarized all the Greek knowledge about geometry and created thereby a new scientific discipline. A special part of his work related the geometric (=earth-measure) which dealt with points, straight lines, planes, angels, distances, etc.. The “elements” of Euclid formed the basis for the teaching of geometry until the 20 Century. The system of Euclid counts of three parts: Explanations/Definitions, i.e. *“lines are parallel if they lie in the same plane, and are the same distances apart over their entire lengths”*. – postulates (axioms), i.e.* “It is possible to draw a straight line from any point to any other point”. - *and common notions, i.e*. “The whole is greater than the part”.*

* *

Euclid put in words five postulates (axioms) of which four postulates were much simpler formulated than the fifth postulate, the so called “Parallel postulate (axiom)”, it states, that in two-dimensional geometry…. “*if a line segment intersects two straight lines forming two interior angles on the same side that sum less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum less than two right angles.”*

* *

For the development of the Non-Euclidian geometry the concept of space, which determined the theory of Euclid was important. It was based on unlimited, most curved levels (planes), and an in three direction unlimited space. Elements of Euclidian geometry are points, straight lines and in the space planes. Only these elements, perhaps even circles and globes are part of Euclidian geometry. The Euclidian geometry influenced the worldview of a disc, which was the only view of the world, which was authorized by clerical and secular sovereigns over thousand of years. Until the 19 Century the Euclidian geometry and his concept of space have been taught. This is mainly the result of the high educational value of the work of Euclid. Even Immanuel Kant affirmed in his works the opinion of Euclid about the spatial structure.

In the course of more than 2,300 years history of the theory of Euclid it was not possible to derive the parallel axiom from the other four axioms logically. The result of the efforts led to new knowledge namely the Non-Euclidian geometry which is just as consistent as the Euclidian geometry but in which the parallel axiom is not valid. János Bolyai wrote about it: *“The way which I followed deserves the more a closer look because the content is important enough and moreover it should be made apparent, that this is the only path which leads to the conquer of the fortress……I have immediately recognized, if the radius (ca) *→ ∞ *is , then the circumference has a limit line in space or, if thought in a broader sense, I have found the existence of circumferences with infinite radius which have the same relationship as circumferences with finitely radius and equidistance straight lines to straight lines like parables to ellipses and hyperbols have. This is unique, I never dropped this scientific result and I lively felt that I found the right way”.*

The research work about the Euclidian geometry was not limited on one place in Europe. Rather, there were three great mathematicians who independently of each other made research work on Non-Euclidian geometry.

In the today Germany it was Carl Friedrich Gauß (1777-1855), in Russia Nikolai Ivanovitsch Lobatschevski (1792-1856) and in Hungary it was János Bolyai (1802-1860) who researched on Non-Euclidian geometry in which is considered: *“For any infinite straight line “L” and any point “P” not on it, there are many other infinitely extending straight lines that passes through “P” and which do not intersect “L” or there exist two lines parallel to a given point not on the line”. *Today, the view has prevailed that the Non-Euclidian geometry is not an anti-Euclidian geometry but it is like the Euclidian geometry a specialization of the absolute geometry, in which the parallel axiom does not apply.

It would go too far to explain the works about Non-Euclidian geometry, which the three important mathematicians Gauß, Lobatschevski and Bolyai independently developed, because for a complete understanding it is necessary to have a deeper knowledge in mathematics and geometry. Due to the limited number of pages the author knowingly avoided a complete presentation of the works of these mathematicians. The development of the Non-Euclidian geometry was very important for the development of new science disciplines, i.e. the cosmology and the general theory of relativity which differ highly from the Euclidian geometry because gravitational fields in space are curved. Out of the three founding fathers of Non-Euclidian geometry it is János Bolyai who combined the Euclidian geometry with all the Non-Euclidian geometries to an absolute geometry of space, while Gauß and Lobatschevski only contrasted and didn’t combine all the theories. After this “masterpiece”, János Bolyai can justifiably be seen as the Mozart of mathematics (geometry).

* *

Since the development of the Non-Euclidian geometry, the question of other conceivable worlds with altered physics and mathematics has bothered many scientists. Kurd Laßwitz (1848-1910) who was one of the founding fathers of science fiction studied mathematics and physics at University in Wroclaw and Berlin, we owe the play *“Prost: der Faust-Tragödie (-n) ter Teil [Prost: Faustus-Tragedy (-n) part]*, in which Mephisto wants to make a freshman the study of Non-Euclidian geometry tasty. The following abstract reflects the conversation between the Fuchs (Fox) with Mephisto:

*Fox: What should I study now?*

*Mephisto: You may try it with analytical geometry. In this study the space is trained, constructed in coordinates, so that, with some luck, from the whole figure, you can get a small part.*

*Then sometimes you will be taught,*

*that what you otherwise impact on one blow*

*constructed in space completely free,*

*an equation is necessary therefore now.*

*Although it was given to the men for their enrichment*

*the three-dimensional spatial intuition,*

*that we can see what happen around us,*

*and that we can construct all the figures-*

*but the analyst enters and proves that it could be otherwise.*

*Equations, which stand on the papers, should be also seen in the space, and if it is not possible to construct them, then we had to define them in another way.*

*Therefore in the infinitely distant two imaginary points all circles must neatly intersect, also parallels, which had to meet, and besides that in the space we can see various Gauss curvature. Because, what we form due to the law of numbers should also delighted us geometrically.*

* *

*The formulars are all true and nice to see; why should they not be able to interpret?*

*There, the students of all places praise that the straight line has become curved. The geometry which is called Non-Euclidian mocks itself.*

**The annual ceremony for Bolyai János at Stifts-Church and Stifts-Barracks**

Every year in mid-November the János Bolyai Honvéd Society together with the National Defence Academy organized a memorial ceremony for the former cadet János Bolyai of Imperial-Royal military engineer academy. The ceremony begins with a mass at Stifts-Church which is celebrated by Hungarian and Austrian pastors and priests. After the mass, Hungarian and Austrian representatives lay a wreath at the memorial for Jànos Bolyai in the Stifts-Church. The ceremony was flanked by guard soldiers, and musicians from the military band of the Gardebattalion of Austrian armed forces form the musical accompaniment to the ceremony. The ceremony is closed by a reception, alternating organized by National Defence Academy and Hungarian Embassy in Vienna. At the reception the leading representatives of Hungary and Austria give a speech to honour János Bolyai.

**Memorial sites for János Bolyai in Hungary and Romania**

János Bolyai is a great son of former Greater Hungary. Most of his life he spent in a part of former Greater Hungary which is nowadays part of Romania. His heritage therefore is kept alive in Hungary and Romania. In his place of birth in Kolozsvár (Klausenburg/Cluj) is still existing the renovated birth house and a monument. Furthermore the “Babes (=Romanian biologist)-Bolyai” University in Cluj and the János Bolyai Institute of mathematics and the University in Szeged were named after János Bolyai.

In some steps of his working memorial plaques exist, i.e. at National Defence Academy, the Stiftschurch and in Olomouc. Furthermore the main-belt asteroid 1441 is named after Bolyai. In today Târgu Mureș** **we can see several memorabilia to honour János Bolyai, i.e. a memorial made by the artists Marton Izsák and Istvan Csorvassy in 1957, and a bust in front of his house, which stands at the crossing of Körösi-Csorna-Sándor-Street. Furthermore it exist an international price of mathematics, which is named after János Bolyai, and which was offered every year by the Hungarian academy of science.

Instead of finding words for honour or a conclusion the author bows to the great scientist and concludes the easy with the suitable statement:

* *

*“With his scientific work to the Non-Euclidian geometry, János Bolyai has initiated the most far-reaching scientific revolution since Copernicus by creating a new world out of nothing. His appendix will stay remain to his everlasting memory.*

“I have created a new world out of nothing!”