Farkas and János Bolyai

Abstract

The lives and works of Farkas and János Bolyai are presented.

Written on the ground of the paper: András Prékopa, “The revolution of János Bolyai”, in “Non–Euclidean Geometry, János Bolyai Memorial Volume” (Eds. András Prékopa and Emil Molnár), Springer, 2006.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Euclid lived in Alexandria, around 300 B. C. There appeared his famous 13 Volume work entitled “Elements”, which has been the source of geometric knowledge for over 2000 years. Also, it has been the model of mathematical reasoning, because it was based on a few given propositions; the theorems were generated deductively from them, all built one on another.

However, one objection appeared from the very beginning. It concerned Euclid’s fifth postulate, that is the axiom of parallelism. Euclid’s definition of parallel lines is the following:

“Parallel lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another.”

The twenty eighth Euclid’s proposition reads:

If the angles between the lines and a transversal line are equal, then they are parallel.

The twenty ninth proposition reads:

The angles between parallel lines (that is the lines which do not intersect) and a transversal line are equal.

To prove it, Euclid used the following postulate:

Euclid’s fifth postulate. If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

We note at once that this postulate is complicated and looks like a theorem rather than a postulate, while the other postulates were utterly simple and therefore could easily be defended as self-evident. Next, Euclid did not used the fifth postulate until the proof of his twenty ninth proposition. Furthermore, the twenty ninth proposition was simply the converse of the twenty eighth one, which was proved without the fifth postulate. That so much could be proved without it, and that, when finally called upon, its use was relatively minor, further contributed to the perception (observation) that the fifth postulate was unsatisfactory and probably unnecessary. In such a way arose the problem of the fifth postulate: for centuries men tried to derive it from the four simple, more clearly self-evident postulates which preceded it. However, their attempts always failed, and, in the best case, gave only its equivalent reformulations.

Among them, we mention only three.

• (1) Playfair axiom (Playfair, 1748–1819). In the plane there is one and only one line that goes through a given point A and is parallel to a given line a, where A is not located on a.

• (2) The sum of the angles of the triangle equals to two right angles.

• (3) Three points that are not on the line are on a circle.

The problem of Euclid’s fifth postulate was resolved around 1830, by János Bolyai and N.I. Lobachevsky. They discovered, independently from each other, a new geometry, the first non-Euclidean geometry, now known as the Bolyai–Lobachevsky geometry, or the hyperbolic geometry.

2 Farkas Bolyai

The influence of János’ father, Farkas Bolyai, to his son’s life was important. Thus, we start with Farkas’ life.

Farkas Bolyai was born in 1775 in Bolya. He came from a Hungarian family of ancient lineage. The fortified castle of Bolya was given to the family in the early 14th century, but in the first half of the 17th century, another János Bolyai lost the castle while was he held captive in Turkey. The family became more and more impoverished and Farkas’ father inherited only a small estate near Bolya. A small estate, close to Domáld, a village near Marosvásárhely, which comprised the heritage of the mother of Farkas, was added to their wealth. When Farkas was about thirteen, he was hired by Baron Kemény as a fellow-student to his son Simon Kemény. This position assured the costs of living and the possibility of education.

Starting from 1790, Farkas and Simon had studied together for five years in the Calvinist school of Kolozsvár. In the fall of 1795, they went to Göttingen to continue their studies. In Göttingen, Farkas made a lifelong friendship with K.F. Gauss, who was a student at Göttingen at that time too. They spent many hours discussing various mathematical problems and, we suppose, the problem of Euclid’s fifth postulate, one of the famous problems of so long time.

After the years at Göttingen, Farkas went to Kolozsvár in 1799, where he was a family tutor for a short time. There, he also married. In 1804, Farkas accepted the position of a professor at the Calvinist College at Marosvásárhely (now Turgumaroş), where he taught mathematics, physics and chemistry. He held this position until his retirement in 1851.

Two children were born from Farkas’ Bolyai first marriage, János and a daughter who died in early childhood. János’ mother was neurotic: there were signs of the problem already in the first year of the marriage, and they grew worse after 1817. She died in 1821, after long suffering.

In Marosvásárhely a solid and spacious house was built for Farkas Bolyai. It was destroyed in 1909.

Fig. 1

The house in Marosvásárhely

The house in Kolozsvár, where János was born, is still there (Fig. 1).

Farkas Bolyai was a very talented man. He devoted his life to prove Euclid’s fifth postulate. Due to him, we have the above mentioned Theorem 3, equivalent to Euclid’s fifth postulate. His main work is the two volume “Tentament”, published in 1832 / 33. It was an outstanding summary on mathematics of that time. Gauss, too, spoke of this work highly, pointing out the author’s precise way of discussion.

Farkas Bolyai was elected as a corresponding member of the Learned Society (former name of the Hungarian Academy of Sciences). The basis for the election was the book entitled the “Elements of Arithmetics”, published in Hungarian in 1830.

Farkas Bolyai was not only a very talented mathematician, but also a many-sided genius. Due to his plays, he acquired a place in the history of Hungarian literature. Another favorite pastime of him was designing the stoves and ovens. He invented stoves of different types and had them made or he himself built them. So the Bolyai’s stoves came in fashion in Transylvania. He gave private music lessons and delivered lectures on the theory of music as well. In addition to Hungarian, he spoke fluently German, Latin and Romanian. He wrote one of the first books on the forestry in Hungarian (Fig. 2).

Fig. 2

The house where János was born

Fig. 3

Farkas Bolyai

Farkas Bolyai died in 1856. According to his will, at his funeral was no other ceremony but “the ringing of the school bell”. Also, his grave was unmarked: only an apple tree of special sort he introduced in his homeland, was planted on it (Fig. 3).

3 János Bolyai

János Bolyai was born on December 15, 1802 in Kolozsvár, in the house that had belonged to his mother’s family. The genius of János already manifested itself in his childhood. When he was six, he learned to read almost alone. A year later, he had learnt German and to play a violin. He was nine years old when his father began to teach him mathematics; at 14, he was very well versed in higher mathematics and worked with differential and integral calculus easily and skillfully. At the same time, János made remarkable progress in playing violin; he already played difficult concert pieces. At 12, he became a regular student at the College. He passed his final exam in June 1817.

Farkas wanted to send János to Göttingen to study mathematics. But the university education was very expensive and the costs exceeded the income of the family. Besides, at that time many of the students at German universities led lost lives: drinking, duels and irresponsible behavior were in fashion at universities, and János was only 15 years old. Farkas was just aware of that and perhaps this is why he wanted János to stay in Gauss’ house. Farkas wrote a letter to Gauss in which he asked Gauss to let his son stay in his house for three years, and offered to pay him for his expenses. But after this request, he destroyed everything when he asked Gauss to answer the following questions sincerely.

“1) Have you not a daughter who may turn out to be dangerous?”

“2) Are you healthy and not poor? Are you satisfied and not grumbling?”

“3) And, primarily, is your wife exceptional among women? Is she not more changeable than a weather-vane? Is she not unpredictable just like the change of barometer?”

We must understand Farkas, especially having in mind the condition of the health of János’ mother. But also, we must understand why Gauss did not reply to this letter.

After this, the possibility that János would study at the Vienna Academy of Military Engineers came up. Farkas succeeded to find some benefactors to provide the necessary money for János’ education in Vienna. After he passed the admission examination, János was registered in the fourth of the eight-year program of the Academy (Fig. 4).

Fig. 4

János Bolyai

During the years at the Academy, from 1820 on, János had been concerned intensively with Euclid’s fifth postulate, i.e. he wanted to prove it. In his letter of April 4, 1820, Farkas warned his son against doing that:

“You must not attempt this approach to parallels. I know this way to its very end. I had traversed this bottomless night, which extinguished all light and joy of my life. For God’s sake! I entreat you leave parallels alone, abhor them like indecent talk, they may deprive you (just like me) from your time, health, tranquility and the happiness of your life.”

János Bolyai finished his studies in 1822, but he was permitted to stay there to pursue further studies for one more year, as he was the best student. In September of 1823 he was nominated sub-lieutenant and was assigned to Temisvár. From there, he wrote to his father his letter, which became widely known:

“My dear Father! Now I cannot say anything else; from nothing I have created a new different world. All other things that I have sent to you are just a house of cards compared to a tower. I am determined to publish a work on parallels as soon as having arranged and prepared it.”

In 1826, János was transferred to Arad, in 1831–to Lemberg and in 1832–to Olmütz. In Arad, János had recurrent malaria fever. Later, he suffered from cholera, too. His health became very bad. This was aggravated by the fact that on his way to Olmütz, his coach turned over and he suffered a serious head injury. In 1833, he was discharged with a pension as a second-class captain.

We already said that Volume 1 of Farkas’ “Tentament”was published in 1832. It was bound together with “Appendix”, written also in Latin, in which János exposed systematically the new geometry, his “new world”, the first non-Euclidean geometry.

Farkas Bolyai sent Gauss a copy, almost immediately after its publication, asking Gauss for his opinion. Gauss’ reply is widely known, too.

“Regarding your son’s work, 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 exactly with my own ideas I have been developing for thirty to thirty five years.”

After the retirement, János lived on the estate of Domáld. From 1834 he cohabited with Rosália Orban. Legal marriage was out of the question because they were unable to raise the money for a deposit that was required as János was an Imperial army officer. They had two children. In 1852, János Bolyai moved away from his family. He was in bad health and taken care of by a servant, which on January 1860, wrote a letter to his half brother: “While I was writing this letter, he died, thus there is nothing to be said: the Captain is gone.”

4 Appendix

The “Appendix” is presented in three fundamental stages. In the first one, János Bolyai rejects the Euclid’s fifth postulate and examines the consequences.

Consider a line a and a point A outside a. Let a line p passing through A intersects a. Moving the intersection in one direction in to infinity, there will be a limiting position, where p does not intersects a any more. We can do the same in the other direction, and obtain two lines both of which are parallel to a. If they are distinct, then there are infinitely many lines between the two that do not intersect a. Each of them is said to be over–parallel with a, and each of them have a common perpendicular with a.

The geometry corresponding to this case is called hyperbolic geometry.

The parallels at a point A determine with normal $AA^{'}$, two congruent angles. This angle is called the angle of parallelism for the segment $[AA^{'}]$. To any acute angle corresponds exactly one segment of parallelism, and conversely, for any segment there is the corresponding acute angle of parallelism. The greater segment corresponds to the smaller angle, and conversely, such that if the segment is sufficiently small, the corresponding angle is very near the right angle. But, if the angle of parallelism is a right angle, the geometry is Euclidean. In short, Euclidean geometry is the limiting case of the hyperbolic geometry, the limiting case when the distances are sufficiently small. This functional dependence of the segments and angles has the following consequence:

If the angles of two triangles are congruent, then the sides are congruent, too.

In other words, there is no similarity in the hyperbolic geometry.

The following and the most important stage in the development of the hyperbolic geometry is the discovery of the horocycle and the horosphere.

Let us consider a set of parallel lines. They have a common point at infinity, say $P_{\infty }$. On lines a and b of the set, we take points A and B, respectively. If the angle $\angle P_{\infty }AB$ is equal to the angle $\angle P_{\infty }BA$, the points A, B are called isogonal corresponding, in short-corresponding points. This relation is an equivalence relation. And each equivalence class is called the horocycle.

The definition of the horosphere in the space is similar.

If instead of parallel lines we consider lines meting in one point, we obtain the circle in the plane and the sphere in the space. Finally, the equivalence class of corresponding points in the set of lines orthogonal to the same line p is the equidistant line, and in the set of lines orthogonal to the same plane is the equidistant surface. The horocycle and the horosphere can be regarded as a circle and a sphere, respectively, of infinite radius. And the line and the plane can be regarded as a limiting case of the equidistant line and equidistant surface, respectively, when the height $[AA^{'}]$ is sufficiently small.

Both Bolyai and Lobachevsky demonstrated that on a horosphere the Euclidean plane geometry is valid, the horocycles having the role of the straight lines. Thus, although the Euclidean geometry is rejected at the beginning, it was not lost; only instead to be realized on the plane, it is realized on the horosphere. This means, that on a horosphere, all Euclidean theorems, relations and formulas are valid, among them Euclidean trigonometry. Using it, Bolyai and Lobachevsky constructed the hyperbolic trigonometry. And this was the third stage in the construction of the hyperbolic geometry by its discoverers.

Investigating the horosphere, Bolyai discovered a so-called universal constant. The change of this constant changes the space, more precisely, changes the metric of the space. This constant remained unknown to Lobachevsky.

5 Recognition of Bolyai–Lobachevsky Geometry

The revolutionary work of Bolyai was not recognized during his life. Also, until Gauss’ death in 1855, the scientific world had not paid any attention to Lobachevsky’s work, although he published his results in German in 1840. Gauss praised Lobachevsky only in letters and not in publications. Nevertheless, as a token of his esteem, Gauss had elected Lobachevsky as a corresponding member of the Royal Society of Göttingen in 1842. However, in Russia, Lobachevsky, who was a professor and rector of University of Kazan, was pushed to the side, because of his “scandalous” theory. Among Gauss’ papers after his death, were found the works of Bolyai and Lobachevsky as well as Gauss’ sketches on hyperbolic geometry. This and the examinations of his correspondence showed Gauss’ positive opinion on the subject, and the works of Bolyai and Lobachevsky were translated in German, French and English. Yet, this did not rise any attention to the matter.

Bolyai and Lobachevsky discovered and systematically developed hyperbolic geometry. However, they had not proved that such geometry existed.

The possibility of the existence of the new geometry was first proved by E. Beltrami in 1868. He examined the differential geometry of the pseudosphere, the surface of revolution of a tractrix.

The tractrix is a curve having the property that the section between the point of contact and the axis of the tangent drawn to any point of the curve is of the same length. The pseudosphere is a surface of constant but negative curvature. Beltrami proved that the geometry of the pseudosphere is locally a hyperbolic geometry. Locally, because the pseudosphere has a line of singular points.

This discovery was a great surprise for the scientific community, and many scholars started to look for surfaces whose geometry is globally hyperbolic. Finally, D. Hilbert proved that such a complete analytical surface does not exist in $\mathbb {E}^3$. But this does not mean that we cannot find a model of the hyperbolic geometry. Such models exist, even in the Euclidean plane.

Let us consider an open Euclidean half-plane with boundary p, i.e. we consider only the points of this half-plane, while the points of the other half-plane as well as the points of the boundary line p are excluded from the considerations. We say that such points are H-points. An H-line is a semi-circle of the considered half-plane whose center is on the line p, or an Euclidean half-line orthogonal to p. Then any two H-points determine one and only one H-line. If the congruence is represented by the product of Euclidean inversions with respect to H-lines, all axioms of congruence are satisfied. (If a H-ine is an Euclidean half-line, the inversion is the Euclidian symmetry). Now, let a be a H-line and A a H-point which does not lie on a. In Euclidean geometry a is a semi-circle meeting p at points P and Q. Points A and P determine a circle $a_1$, while A and Q determine a circle $a_2$ whose centers are on the boundary line p. But $a_1$ and $a_2$ are H-lines, and because P and Q are not H-points, they both are parallel to a H-line a. And this means that in the considered model, the hyperbolic axiom of parallelism is valid. Thus, the model of hyperbolic geometry is obtained. It is called Poincaré’s half-plane model.

Thus, is there something wrong in the hyperbolic geometry, it will manifest itself in any model of it and therefore in Poincaré’s model, too. But this model is a part of the Euclidean geometry, that is we would have a contradiction in the Euclidean geometry. Thus, if we suppose that the Euclidean geometry is without contradictions, the hyperbolic geometry is without contradictions, too. Conversely, if we suppose that the hyperbolic geometry is without contradictions, so is the Euclidean geometry, because one model of the Euclidean geometry is the geometry on the horosphere, which is a part of hyperbolic geometry. So, these two geometries have equal rights.

In the other model, called Poincaré’s disc, the points are those of the open circle (that is without boundary) and the lines are the circular arcs within the open circle such that they meet the boundary perpendicularly, as well as Euclidean straight lines goings through the center of the boundary circle. As before, the congruence is represented by the product of inversions, and the hyperbolic axiom of parallelism is valid.

In a similar way, we can construct the model of hyperbolic space in the Euclidean (open) half-space, or inside the open sphere.

In 1872, F. Klein constructed comparative projective models of Euclidean, elliptic and hyperbolic geometry.

Klein’s mode of the hyperbolic geometry is the open interior of the real, non-degenerate conic k. If the projective line a intersects k at P and Q, the (open) segment PQ represents a hyperbolic line. Thus, there are two lines that go through a given point A and are parallel to lines a : AP and AQ. The line b intersecting a at a point S outside k is over-parallel to a; their common normal is the line LM, where L is the pole of a and M is the pole of b. As for the group of the congruent transformations, it is the subgroup of the projective group which does not change the boundary conic k.

Later, this outline was completed with other non-Euclidean geometries. In the plane, there are 9 such geometries and in the space-27. Between them there is the pseudo-Euclidean geometry. (It is recalled that the geometry of the special theory of relativity is pseudo-Euclidean geometry of four dimensions).

Remark

D. Balnuša [1] proved in 1955 that there exists in $\mathbb {E}^6$ a surface of constant negative curvature, and E. R. Rosendorn [6] in 1960 proved the same for $\mathbb {E}^5$.

6 Applications

Hyperbolic geometry has many applications both in mathematics and in other sciences. In mathematics it is a very useful tool in the theory of complex functions. When Poincaré considered (1881) complex functional linear transformations with real coefficients, he noticed that they preserve the complex upper half-plane and that this half-plane provides a model of the hyperbolic plane.

A. Sommerfeld established (1909) the connections between the formula for addition of velocities in the theory of relativity and the trigonometric formulas for the hyperbolic functions, and V. Varičak [7] showed in 1912 that Sommerfeld’s formulas are formulas of hyperbolic geometry. Using hyperbolic geometry, Varičak also interpreted the Lorentz transformations, Einstein’s formulas for the aberration and Doppler’s effect. Hyperbolic three-dimensional space has been applied to problems of relativistic physics by N.A. Černikov [2] and to relativistic kinetics by Y.A. Smorodinskii [8].

There are applications in arts, too. The most interesting is the artist M.K. Escher. Having studied the figures of hyperbolic geometry of H S.M. Coxeter, Escher created, in 1959 and in 1960, his marvelous “circle limit” engravings, in which we recognize at once the Poincaré’s disc model of hyperbolic plane and H-lines.

*

* *

By 1883, when he addressed the British Association for the Advancement of Sciences, A. Cayley discussed the question of non-Euclidean geometry as an unimportant: “My own view is that Euclid’s twelfth axiom in Playfair’s form of it does not need demonstration, but is a part of our notion of the space, of the physical space of our experience - the space, that is, which we became acquainted with by experience but which is the representation lying at the foundation of all external experience.”

But the most important consequences of the discovery of the first non-Euclidean geometry (supplemented by Riemann’s approach) is just the new interpretation of the geometry and the physical space, radically different from Cayley’s view that the world is Euclidean.