Lukács B.

CRIP RMKI, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

lukacs@rmki.kfki.hu

For commemorate the bicentennary of the birth of Bolyai János, construer of non-Euclidean geometry, I write some observations, notes, explanations &c. which generally do not appear in international literature. Also, I want to calm down his shadow, maybe as reckless and angry as he was in life.

According to Hungarian traditions and State regulations, also to common sense, family names come first. Also, note that Hungarian and Magyar are not synonymes. Further details are in the text.

[X] stands for Note if X is a capital Latin letter, and a Reference if X is a number. In citations of form [@X] the material is an unpublished packet of numbering X of the Bolyai manuscripts in the Teleki Téka, and K x/y is a numbered Bolyai manuscript in the Manuscriptorium of the Hungarian Academy of Sciences, Budapest.

Transylvanian localities have multilingual names. In Middle Ages Latin was the official one, if existed.

The very specific Hungarian letters which are the long counterparts of ö and ü are written here as ô and û. While this is incorrect, it is full proof on HTML, Hungarians generally can recognise the trick and non-Hungarians do not know the original letters at all.

Although international priority arguments are multitudinous, I think, the Hungarian Academy of Sciences can prove that Bolyai [A] János has priority as the person constructing and publishing the first true and rigorous non-Euclidean geometry [1]. While Gauss may have known a lot (or may not), he never published (although clearly he had the mathematical apparatus at least in 2 dimensions used later by Riemann in 3) and the German publication of Lobachevskiy is clearly second [2]. True, there is a Russian Lobachevskii work.

But I am no historian, so for the 200th anniversary I do not discuss priority in details. Some notes will be made in Chapter 6 and Appendix B. I will not discuss the ways to overcome the necessity of the Euclidean postulate/axiom of 2 right angles at all. I want purely to commemorate Bolyai János.

In his life he could not discuss the parallels with anybody; and in his last 25 years it seems that he could speak practically only with his father and his sweetheart; with both of them shouting, denouncing &c. It must have been a not quite satisfactory existence. I think that around the 200th birthday his shadow is happy if we discuss him anyway at all. So I will discuss his works & life from angles not taken by anybody else.

The answer is Yes in abstract sense. Bolyai János has shown that the Euclidean axiom of parallels can be substituted with another, and the construction remains OK. So as a mathematical possibility/virtuality it exists. However has Physics found it anywhere?

It is a rumour that Bolyai János believed that the 3 dimensional space of astronomy has a Bolyai geometry, with an angle deficit somehow connected to densities of astronomical bodies or such. If so, this was an erroneous guess: the space of astronomy is a space-time: four-dimensional and pseudo-Riemannian. Bolyai would have refused such a construction.

But there is a 3 dimensional space with positive signature which is approximately of Bolyai geometry: the (central part of the) space of male human colours.

Vast quantity of distinguishable colours exist. But even qualitative observations strongly suggest that colour space of normal males is 3 dimensional.

First, it seems that any possible colour can be described by 3 data. First, the absolute brightness or albedo of the surface. Second, the dominant hue. This is either the colour of an ideal monochromatic reflectant, or, loosely speaking, of a monochromatic spectral light; or, in the purple, a mixture of far red and blue, with some relative mixing. Then, this colour is of 100 % saturation; one can get colours of less saturation by adding neutral white in the needed quantity. Indeed, as early as in 1947, van der Akker & al. [3] demonstrated that this mixing is viable. It seems that no colour is left out.

Another confirmation comes from the 3-colour theorem of Maxwell. It states that any human colour impression can be reproduced by means of appropriate mixtures of 3 fixed coloured lights. Obviously the theorem cannot be true in face value, because no mixture can be more saturated than the original lights, so problems arise at very "pure" colours. Also, now there are reports about tetrachromate women. Finally, if rods influence colour impression, then moderate tetrachromacy appears even at males [4]. Still, trichromacy can be accepted as first approach. Colour photography, colour TV, colour printing and computer monitors are based on the theorem.

Now, colours are reproduced via colorimeters: optical apparatuses mixing the lights of 3 coloured lamps. Tens of thousands of hues were identified this way [4] with the tolerance ellipsoids around. Now, if per definitionem, the distance of 2 just distinguishable colours is unity, then the metric tensor of the colour space of the average (male) observer is known in more than 10,000 points. And this metric is not Euclidean [4], [5].

One dimension is almost trivial: the neutral amplification, the simple intensity. Two lights of the same hue but different intensity are distinguishable, but the property in which they differ is not quite colour. Now if we use lights of the same brilliance, luminance or such, we are on a surface. Which surface?

Instead of direct answer, it is useful to follow Ref. [5]. What is the purpose of colour vision? Most probably identification of surfaces. So clearly, eye & brain wants to measure the reflectivity, not the reflected light. The latter would depend on the illumination too; and human "colour constancy" is quite good until the illumination is not absurd.

But then look.

(2.1) V(l ) = R(l )*I(l )

where V, R and I are the light, reflectivity and illumination, respectively, and the eye observes 2*N linear functionals of V and I; N is cca. 3. The brain cannot hope to reconstruct more than N data of R(l ); and it can do for N only approximately. The approximation is most self-consistent if the brain assumes functional forms invariant under multiplication/division, so Weinberg turned to forms

(2.2) (V,R,I)(x) = exp{a + bf(x) + ch(x)}

where f(x) and h(x) are some good assumptions, fixed. All results of multiplications/divisions remain in this class, only the coefficients (a,b,c) change.

Hence he was able to get a metric. If

(2.3) f(x)=x; h(x)=x²

so the assumed intensity distributions are Gaussian, then the space is not Euclidean.

Now, many years ago I detected a minor point where the Weinberg theory needs a generalisation [6]. The brain may have any assumptions; but for innumerable millenia did not know anything about wavelengths. Now, take distributions of form (2.2) which are Gaussian if x is the l wavelength; their forms will not be even of (2.2) if x is frequency n = c/l . so in V and I a multiplicator must appear

(2.2’) (V,I)(x) = W(x)*exp{a + bf(x) + ch(x)}

to guarantee transformation properties; but not in R. But this is quite fortunate, because natural illuminations are rather similar to solar spectrum, so V(x) may be the noon solar light or some similar. Solar illumination is not Gaussian (I do not go here into arguments why a Gaussian form is physically expected, approximately; see [5]), rather a Planck spectrum. But the brain can put this into W(x). (And Weinberg Theory contains one more function, C(x), the luminous efficiency, about which simply nothing will be told here; if somebody is interested, he can consult with Refs. [5] and [6].)

Now, assume that everything is Gaussian, even W(l ). (For W the best fit seems to be <x>=500 mm, s =360 mm.) Then again everything can be analytically calculated. Details can be seen in Ref. [6]; here is only the essence. Colours of fixed luminance are of course on a surface and the simplest coordinates are (X,Y), where X and Y is defined by the physical parameters of a light generating the respective colour, as

(2.4) X = (l_oS²±ls²)/Q

(2.5) Y² = s²S²/2Q

(2.6) Q º S 2 ± s 2

where the upper sign holds for "normal" hues, the lower for purples. (A lot of other, non-Gaussian, distributions produce the same colour, of course.)

Now, the metric on the surface turns out to be

(2.7) ds² = q²{(dX² + dY²)/2Y²}

where factor q is responsible for individual experience, so q is large for wall painters &c. Take for simplicity q=1; it is irrelevant in the geometry.

Geodetics/straightest lines on this surface are i) vertical lines, ii) half circles with center at Y=0. The spectral "loop" Y=0 is the lower bound of the surface; the upper one at infinite Y is some purple (and the model has a singularity at saturated purples, because of manifold simplifications). But for distance Y=0 is also infinitely far from the other colours.

Let us see first a vertical line. It starts from somewhere on the spectral loop. X is constant. Going upward the dominant hue is kept but more and more white is added. At Y=34 mm we go into purples, first very unsaturated then purer.

Now the half circles. A half circle has two endpoints at Y=0, so on the spectral loop. If the hues are not too far, we get a familiar chord of the loop: the most straightforward path between bluish green and yellowish green runs through not too saturated greens &c. Between say oranges and turquoises the most straightforward route goes through purples, so the half circle reaches higher than Y = 34 mm.

Now, this surface has no preferred point anywhere; it is easy (for a non-Euclidean geometer) to show that it is maximally symmetric too. In fact, it is Poincaré's half-plane picture for the Bolyai plane [7], [6].

But if somebody does not directly see this, think qualitatively. The equivalents of straight lines are the shortest lines or geodetics; now the vertical lines and half circles. Take 2 verticals and a half circle just between. Parallels meet in the infinity (so either on the spectral loop or in saturated purple), ultraparallels even not there, and "familiar ones" in the finite. Now the half circle and any of the two verticals are parallels, meeting in infinity, at Y=0, and the two verticals are ultraparallels to each other.

The approximation is not really good in infinity; the simplified model is not too good for real human colour vision for very mono- or bichromatic lines. But it is quite good in moderate finite. E.g. form (2.3) seems true from bluish green to almost pure red.

Why human colour vision produces an approximate Bolyai geometry in the colours in brain? I try to answer in two halves. First, there are no preferred points & directions on a 2 dimensional Bolyai surface. Now, reflectivities are so various around us that it is not surprising that Evolution prefers to handle all of them similarly. Only Sun itself could define a preferred distribution ("colour of Sun"); but Sun's data just drop out in the formalism because of colour constancy [6]. So Evolution is behind the constant curvature of the surface.

But why constant negative curvature? OK, I do not want to have copyright argumentations. In the autumn I will write my lecture held at the Agropolis Bolyai conference [8] and Acta Physica Hungarica plans to publish the lectures. So, if you will be still very interested next year...

The Latin term is more appropriate: Doctrina salus. But it was written in Magyar (although not with the official orthography & letters) and remained in manuscript. The amount is cca. 60 handwritten books, unbound. The idea is the welfare (salus) of Humanity by means of learning (especially geometry, of course), plus extensive social reforms.

It seems that in this moment no European parlamentary party would accept the theory, while it was too European for Khmer Rouge. Also, it is definitely against old Magyar traditions at the Eastern European steppe, and also against the yassak of Gengis Khan, because its idea is to organise the population into groups of 12, not of 10 or 7.

The Salvation Theory is unpublished. In addition, it is not sure that it is complete; after the death of Bolyai János, Ret. Ing. Cpt., the Imperial-Royal [B] Army took all the manuscripts to see if they contain military secrets. After some time they became completely uninterested and transferred the papers to the Agropolis police station where they were kept in a shed. The Agropolis College after some time took over the papers. Therefore, while there are no indications that anybody confused anything, papers may have been lost. Anyway, now the manuscripts are in the Teleki Téka, in Agropolis/Marosvásárhely/Tirgu Mures/&c. and from the 50's historian Benkô Samu read them. He published some details in the 70's [9], but one must remember the ideologic situations in Roumania in that time; so he was very careful. The references are here [@x], where x is the number on the respective ms. packet in the Teleki Téka.

The manuscripts cover a much wider area than the Salvation Theory (drafts of curriculums, drafts of lectures, strictly personal notes (see next Chapter), a study on complex algebra &c., but except for the ms. of Appendix to Tentamen (the Absolute Geometry) all unpublished. It is not always easy to tell if something belongs to the Salvation Theory or not. Now it is told that the Salvation Theory material is cca. 60 volumes; but it is not easy to measure.

The Theory has 3 parts. The Salvation Theory proper (Üdvtan), an Appendix to it (Meltan), and a Historical Part (Multan) which is also called Appendix to the Appendix.

The Salvation Theory proper is divided into 4 parts: Language, Quantification, Aesthetics and Philosophy. The last one means purposes, and ways to reach them.

Some scholars guess that the idea of the Salvation Theory is from 1832, so just after the publication of the non-Euclidean geometry. Namely there is a draft of an application on 3rd May to Archduke John [10] for three-month furlough "to fully elaborate" some non-detailed good ideas. Archduke John refuses the application.

However there may be some misunderstanding. Namely, both the draft and the final application can be found in the Manuscriptorum of the Hungarian Academy of Sciences, as K 23/61-62, and it seems that he applied for 3 years furlough, which is quite another matter [11]. Bolyai considered the refusal and the process unfair; however from Benkô's study [9] we can see that it was fair enough. Let us see what happened in 1832.

Appears the Appendix; the Work. On 29th January Lt-Col. Zimmer sends Bolyai's name for captaincy; he gets the appointment soon. Gauss sends a letter to Bolyai Farkas, mailed on 6th March, congratulating for the Tentamen, and also for János' Appendix. The draft for furlough is 3rd May, but it seems that he sent it finally on 8th Aug. In the application he uses strongly the authority of Gauss.

Archduke John is not a geometer. So the letter and enclosed material goes to Cpt. Greisinger Gustav Adolf, teacher of the Military Engineer Academy. Greisinger answers on 14th Sept., he writes that he cannot see the importance of the Appendix (see K 23/63). Then Archduke John asks the opinion of Freiherr von Ettingshausen Andreas, Professor of the Vienna University. The Freiherr gives some opinion, according to which on the records of Cpt. Bolyai a note appears that Councillor of State Gauss commended a mathematical work of the Captain; which is indeed true. Archduke John refuses the application for three-year furlough. However the captain has some medical problems. Comes 1833, and on 11th Apr. Major Zitta sends the captain to medical reexamination. Probably according to the results, Major Zitta signs the discharge with a pension on 11th June (K 23/64-65).

Maybe if Gauss had written that the Captain had solved a 2000 year problem, the Imperial-Royal army would have given 3 years with full salary. OK, Gauss wrote more cautiously, but the Bolyais knew this. Had the Captain asked for a mere 3 month furlough, he surely would have got it even with the Gauss letter. But he asked for 3 years. In addition he has medical problems: then the Army puts him on pension, he gets a moderate amount of money forever, has nothing more to do, and can work on his ideas.

However I believe that Stäckel [10] is in mistake, and the plans obscurely mentioned to Archduke John were not those for the Salvation Theory. Benkô found a Foreword draft in [@759] explicitly stating that the idea came on the night to 2nd Feb. 1841. In this time, from 1834, Cpt. Ret. Bolyai lives at the farm of his father, on Domáld/Viisoara, with his 2 small children Dénes (1837) & Amália (1840) and the housekeeper kibédi Orbán Róza of Kóródszentmárton (Sedes Maros). We do not know too much about visitors, it is not a bad guess that there were none, and we know that the captain hated the farmer life [@286]. So the only company to discuss the Salvation theory was the housekeeper; but from the content of the theory I can guess her answer if the captain starts to discuss.

The starting point of the Salvation Theory was not new. The Society needs serious reforms. Just now there is no harmony between personal and common interests. The base egotism prevails. The small people is poor and helpless, &c. This can be remedied if private landownership ceases. In the Good Society everybody has to have 2 hour agricultural work daily. That will be good for him/her.

As I told, so far this is not new. One can remember Rousseau; or a lot of French, ad absurdum till Deschamps. The land is common, people are against luxury, they live simply and naturally &c.

Bolyai János is not too fanatic. There will be "common workshops". People go to work on timetables, 6 hours for sleep, 2 hours agriculture, 2 hours in the workshops, 2 hours for eating &c. The crops are kept in common granaries. there will be supervisors on the fields. The reeve is the Chief Supervisor, above 12 secondary supervisors and so on.

He is ambiguous about money. In [@604] he suggests a future without money, but in [@650] he thinks that money is a good tool, only we must be sure that nobody wants to collect it. Commerce remains, but the leaders supervise it that merchants cannot earn more than fair.

Everybody will eat fried or boiled food, not enticing of course, just in the proper quantity; e.g. at noontime half a pound meat in soup, with enough bread "as in the army". In the evening maybe polenta with milk. It would be good to eat together: 12 families [@542]. It is best to build 12 buildings in a street. The idea should be propagated on the whole Earth, which is divided into 12 regions. This propagation of course needs conspiration; in cells of 12. Of course, the first dozens will be selected by the Founder, the Captain.

Children are taken away from the mothers at age 1/2; then they are educated in groups of 12, with no co-education. Education is ready at age 24 (2*12?).

Let us stop here a moment. It would be too simple to tell that after a century a regime introduced common lands, common workshops, regulated commerce and system of Chief of Chief Supervisors (called Conducator), supervisors, sub-supervisors and so on in Bolyai's Agropolis/Marosvásárhely/Neumarkt/Tirgu Mures, Dolmánd/Viisoara and Bólya/Bell/Buia with the result that in 1985 meat simply vanished (with a lot of other foodstuffs) and the women went to the forests to collect mushrooms and snails while an organisation of supervisors tried to confiscate the above goods because the forests are commons so nobody can collect there individually. However Bolyai was not alone with the idea, and surely the Conducator did not read the Salvation Theory.

Also, it is not a big success for the constructor of Earth's first viable non-Euclidean geometry to repeat Rousseau a century later. But Rousseau even now is regarded as a mental giant. And he is not repeating Rousseau. He wants the rapid development of Science. Only the common land and simplicity is common with Rousseau; and I agree with Bolyai that polenta is good to eat. Only, I do not like it with milk; it is much better with soft sheep cheese (bryndza).

However we would expect World's first geometer not to make self-inconsistency/bad logic. See then. He wants to propagate the New Life. Good. He wants to propagate secretly. I am not surprised: maybe the Emperor and great landlords would not like it. Then: why to write a suggestion to Emperor Francis Joseph I of Austria for the betterment of public matters in the spirit of the Salvation Theory? The autograph of the suggestion can be found in the Manuscriptorum of the HAS, under K 24/5 [11], written on 29th July, 1852. But let us continue. I told that the idea, although it is similar to those of French intellectuals in mid-XVIIIth century, has some specialities. Not only the duodecimal system, although scholars would prefer decimal, following the numbers on their fingers, while scientists would like at least duodecimal, maybe hexagesimal (as Sumerians did), because of much more integer divisors.

And look: the Captain is quite cautious in sexual matters, contrary to military traditions. Also, he has already two small children, so he is not a monk. (Anyway, he is Calvinist.) Still, he writes that "...for diminishing the sexual stimuli it would be better to abolish the differences in dress of the two sexes". During education (so up to 24) boys and girls should not even know about each other [@59]. Then comes choice: Men and women, coming just to age, come together naked and choose the other. Then the pairs go away, in groups of 12 [@230], and it seems, remain faithful to each other.

Obviously, the naked youth come from the stories about Spartans. Also, the idea might be good or bad independently whether the Captain (Ret.) followed his own ideas or not. But again: bad logic! You educate 12 boys and somebody else educates 12 girls. They never saw each other. Now they are 24 old, you strip them and herd together. They are going to choose.

Are all girls very similar? The boys will not choose according to anything else than appearance (and maybe odour) not knowing anything of the girls at all. Say, two would like the same girl. Should the boys fight? Or what? The mathematician should prescribe a process, e.g. lottery, dice or such; that would be madness but would work. His suggestion, however, would not work at all. I would overlook such an error at a philosopher, but not at the successful inventor of hyperbolic space.

There is a complete autobiography, again unpublished [@1086]. In it he writes: "Who is worse than snake? Tiger. Than tiger? Daimon. Than daimon? Woman (but only bad woman because the good one is better than angel). Than woman? Nothing". Now, without the bracketed term (his own) it is a 2500 year old cheap joke; the bracketed term is very romantic; what is this, and why? The Captain is 44 old, a veteran, a father of two, and first geometer of the world. Or: is he again in love? No letter is known about. No stories.

Obviously, the captain-geometer does not rely on women. Obviously he did not rely at least for 10 years. During that, by the way, two children were born to him. Obviously in Salvation Theory he tries to diminish sexual temptations. Only I do not see how and why. I return in the next Chapter how he chose the mother of his children; but about the clothes to diminish temptation (same for men and women; and, important, pale apple green): does the actual fashion count much if they are alone and together permanently?

But now let us see the point where Bolyai is unique among utopists. Above I started to cite the ideal timetable. 6 hours sleep, 2 hours on the fields, 2 hours in the common workshop, 2 hours for eating. That is half of the day. And: 12 hours for the Salvation Theory and noble pastime!

Now, after our XXth century I am already not surprised that somebody believes that all people of World should learn and elaborate Ideology. But this is really new in 1850. In the French Republic, when world's best chemist Lavoisier was put on trial (to be sure: not for being chemist, but for being count and chief tax collector) and some academicians tried to intercede on behalf of him as a great scientist, Chief Public Prosecutor M. Fouquer-Thinville replied that La Republique does not need scientists. Dom Deschamps is more radical. Art is unnecessary. Books are unnecessary except for one in which The Programme is written. Therefore children will not learn to read: why at all [12]? Science is luxury. Everybody should always work on the fields. And a lot of other agrarian utopists might be cited, the most recent being Pol Pot.

Now, Bolyai likes Science; and likes some Arts. One of his problems with the present society is that the sciences evolve rather slowly [@636]. As for Arts, the first is Music. However caution is needed because it may lead to the madness of living with the other sex. But Poetics is not good for anything.

As for Sciences, History is needed but only as a memory of the old, wrong world [@292]. The elimination of luxury items will make Society to be able to concentrate on really important tools of Science, namely "good, black (not shining) permanent ink", plus "pale, immature apple green paper" [@723]. Sure, if everybody is amply provided with good ink and apple-green paper, then he (she?) will cultivate The Theory hours daily.

The adolescents, from 12 years, start to learn mathematics/geometry. The primary teacher is an older pupil. But for each 12 there will be a true teacher. And the society in itself is visualized as a scientific body. The citizens will argue on Sciences, even on The Theory. If somebody has a new result, it will first be discussed among the 12, then goes upwards. If it is really good, it will be published.

Puccini's Madama Butterfly is a historic snapshot; but XXth century intellectuals could generalize the idea. Rich and advanced Europe intrudes poor societies. European males seduce/exploit girls of Nature; and then unscrupulously abandon them. The girl is very sad, plus how can she make a honourable life afterward? But the travellers/conquerors are without conscience.

Now, I am almost sure that the topics of Madama Butterfly does not support the message of Madama Butterfly. The story belongs to 1870, maximum 1880. Japan is not poor even then; but still adapting to England, USA and Prussia. Marital mores are still traditional. So, Madama Butterfly may be sad at the end; but can start a new life. She may have been a temporary concubine in her own, undisturbed society too.

But I think, in the XIXth and XXth centuries similar situations may have led to real conflicts, say in India. The rich and powerful colonizer disintegrates local societies, snatches away the nice girls and so.

Now, for first sight this is just done by Captain Bolyai. Let us go step by step.

The father, Bolyai Farkas, is not at home. His ancestral land is the land of Hungarians (Lakság), especially Bólya in County Felsô Fehér. He goes to Agropolis/Marosvásárhely in 1804, when applying there. Agropolis is in Sedes Maros [C], on Szekler land.

OK. Is he a coloniser? We can argue back and forth. For example, for a Szekler some 90 % of Hungary is Hungarian, including the Greek Catholic Bishop of Blaj/Balázsfalva and the King/Emperor Francis I/II in Vienna/Vindoboba/Bécs. (As King of Hungary.) World is full with Hungarians, and they are in average richer than Szeklers. But Bolyai Farkas is neither richer nor more powerful than the Szekler colleagues and leaders of Agropolis [D].

Now let us jump to 1834. The retired Captain agrees with his father, Farkas. The father must live in Agropolis, so János can use the family farm at Domáld (Viisoara), 40 km thence, On Hungarian land. He will not pay, but has to send 2/3 of the winter fruits (especially the pónik apple) to the father plus one mathematical study per year [@815]. (The last will not be fulfilled.) But János needs a housekeeper.

Just before the way to Dolmánd would cross the border river Kis-Küküllô [E] there is the last Szekler village, Kóródszentmárton. There the last building before the border bridge was that of the family kibédi Orbán. They had a girl, Róza, going to be a spinster. The girl agreed to be housekeeper for 7 florins per month. This was in 1834.

Then in 1837 came the baby Dénes. Bolyai János recognised the baby as his, but no marriage. And then again 3 years, and another baby, Amália. And kibédi Orbán Róza still a girl. With two babies. And the kibédi Orbán family esteemed itself at least as high as a real landowner in Hungary, and at home had as much rights (not as much money). The Captain snatched away the daughter of an esteemed local family, carried away into foreign lands and ruined her, did he not?

Well, Yes and No. Yes, because at least the possibilities of Róza to start a honourable family much diminished. And No. It seems that Orbán Róza did not want to start anything new. Sure, the household was not in peace. But the picture we can partially reconstruct was not an unilateral sadness. It was rather an Italian comedy. From the sources one get the impression that Orbán Róza easily overshouted the retired Captain. We can find various papers on which the Captain records (for whom and why?) that Róza is impossible. On the backside of the draft of a will -in which Róza is the sole inheritor!- he writes that Róza is "most impertinent, unenduringly venomous, saucy, rebellious, and vituperates even in others' presence" [@243]. Well. Maybe he is right. And then? Surprisingly enough, he applies to the King/Emperor to get permission to marry wihout caution-money Orbán Róza, "a woman from good family and conduct". Sure: the kibédi Orbán family is good. They are free Szeklers. At least as good a family as that of Hungarian noblemen (say, the Bolyais). And how good is Róza's conduct?

For understanding, Bolyai János is a military officer. He must deposit caution money before marrying. He does not have the money. His father does not want Róza. And the kibédi Orbáns? We do not know; but they do not pay either. There are quarrels. Who is the father of Dénes/Amália? Not you, you can be sure! Then the Captain goes to the church authorities, telling that one child is not his. They make the appropriate record. Then again, he goes and tells that now he is sure, the child is his own.

And in April 1849 the Hungarian Parliament dethronize the Hapsburgs. In five weeks the happy couple is marrying; free of charge. In 4 months the Hapsburgs are back but what can they do?

In 3 more years the happy (?) marriage breaks down. Róza, with the children, goes away. According to historians this is final. And then: after 4 years, Róza produces another child, Gyula. And the Captain recognises the child [@772]!

Who was exactly ruining whose life? (Well, of course, verbally Hungarians are weaker than Szeklers.)

As we know, the genesis of General Relativity went cca. in the following timetable:

1630 Galileo's Relativity Principle is convincing, and can be proven in Mechanics starting from axioms/assumptions which seem obvious (e.g. that forces depend only on distances between bodies and not on absolute positions). In the same time the Galileo Relativity does not hold in Optics.

1875 Marity Mileva born at Titel, Bács-Bodrog County, Hungary.

1878 Grossmann Marcell born in Budapest, Pest-Pilis-Solt-Kiskun County, Hungary. His father, Grossmann Gyula, is founder and part-owner of a big factory producing agricultural machines, at Külsô Váczi út 7 (almost at present Lehel tér).

1879 Einstein Albert born at Ulm, Schwabenland, Germany.

1881 Michelson's interferometry experiments are unable to detect Earth's velocity by optical tools. So either Earth is at absolute rest (Aristotle & Ptolemy), or Galileo's Relativity is synchronous for Mechanics and Optics (somehow), or something more complicated holds. Marity starts her schooling in Titel, Hungary.

1886 Marity in middle school in Sabac, Serbia.

1887 The Marity family goes to Mitrovica, Croatia; Marity in middle school in Zagreb, Croatia

1893 The Grossmann family (of Alsatian origin and strong Swiss connections) settles in Switzerland.

1895 Lorentz suggests a principle: everything moving contracts in the direction of motion by (1-(v/c)²)^1/2. If that is true, then Earth's velocity cannot be detected via interferometry; and if all internal forces are electrodynamic, then the contraction follows, even the formula can be proven in first approximation. However then this Lorentz contraction makes impossible to measure the length of anything, because measuring rods are uncontrollably contracting too. So work continues. Poincaré starts on the problem. Marity in Switzerland for the last year of middle school.

1896 Marity, Grossmann & Einstein start at Eidgenössische Technical Hochschule, Zürich.

1899 Einstein becomes interested in problems of ether, motion of dielectrics &c. and proposes mutual work to Marity (from Love Letter #8 upwards), who, however, seems uninterested in ether [13]. Marity gets the lowest passing grade at geometry exam of Fiedler. Einstein had that exam in 1898, Grossmann specialises to geometry.

1900 Poincaré on principle of equality of action and reaction and on Lorentz contraction. Clock calibrated via light.

1901 Grossmann is working on his dissertation at Fiedler: "On the Metrical Characteristics of Co-Linear Formations". As Einstein writes to Marity (Love Letter #48): the "...subject ... is related to ... non-Euclidean geometry. I don't know exactly what it is.". A work of Marity & Einstein appears with a single author Einstein about thermodynamics.

1902 Marity Liza (daughter of Marity & Einstein) born, given away and vanishing. Grossmann gets his doctorate from non-Euclidean geometry at University of Zurich. Einstein starts at the Patent Office, technical expert, third class.

1903 Marity tries to get two teacher positions at Belgrade Middle School for herself and Einstein. Marity marries Einstein and renamed Einstein Mileva. Coup d'état in Belgrade; Serb military kills the King from the Obrenovich family and crowns the Karagiorgievich pretender.

1904 Poincaré on the verge of the new Relativity Principle.

1905 Annus mirabilis. 4 resounding Einstein articles, including the Electrodynamics of Moving Bodies. Discrepancies between relativity principles in mechanics & optics are eliminated on the cost of so called Lorentz Transformations. Einstein gets his doctorate. Poincaré gets Bolyai Award from the Hungarian Academy of Sciences.

1907 Einstein starts with the problem of "non-inertial motions". He considers the specialisation to uniform motions unnatural, but then the transformations become quite arbitrary.

1908 Minkowski shows that Einstein's Lorentz Transforms are rotations in a 4 dimensional pseudo-Euclidean space-time.

1909 Einstein is Dozent at University of Zurich.

1910 Hilbert gets Bolyai Award from the Hungarian Academy of Sciences.

1911 Einstein is Professor at the German University in Prague.

1912 Eintein is Professor at University of Zurich. Discussions with Grossmann restart. Einstein states that he would need an "absolute calculus" for arbitrary coordinate systems in space-time. Grossmann tells that the absolute calculus is ready in non-Euclidean or Riemann geometry.

1913 Formulae of non-Euclidean geometry are applied on pseudo-non-Euclidean ones called then pseudo-Riemannian. Article of Einstein & Grossmann is published about Generalised Relativity.

1914 Einstein Albert at Berlin; Einstein Mileva does not follow. World War I starts.

1915 Hilbert finds the simplest possible governing equations for the curvature of a non-Euclidean metric space. Hilbert nominates Einstein to Bolyai Award, but the Award is not given because of WWI. (The Hungarian Academy of Sciences offered all her money to the State for the duration of the War.)

1916 Einstein publishes the classic article of General Relativity. He accepts Hilbert Equation as gravitational equation.

1918 Effective end of WWI.

1919 Solar eclipse proves General Relativity. Einstein & Einstein divorce.

1920 Peace of Little Trianon Palace gives Agropolis & Domáld of Bolyai to Roumania and Titel of Marity to SHS Kingdom, later Yugoslavia (effectively Serbia). Grossmann's Budapest remains in Hungary.

1921 Einstein gets Nobel Price for photoelectric effect.

1936 Death of Grossmann in Zurich.

1948 Death of Einstein née Marity in Zurich

1955 Death of Einstein in Princeton.

1969 Marity monograph U senci Alberta Aystaina (In the Shadow of A. E.) by Trbuhovich-Giurich Desanka.

1975 Start of repeated Marcel Grossmann Meetings, organised by Ruffini Remo.

Admittedly the above timeline contains entries of various importance; but this is so because I am preparing a future study about the Relativity Idea [F]. However, the above entries suggest an interesting triangle in which ideas were developing.

Today it is usual to speak about Special Relativity as "Einstein-Maric" theory in female university circles. This is impossible, not only because Marity Mileva did not use the Croatian orthography "Maric" but also because the Love Letters [13] prove that she was not interested in the luminiferous ether problem at least up to 1903 when letters, of course, stop. Some theories about heat transfer and capillarity, in the same time, should be called Einstein-Marity; however Thermodynamics does not belong to the present study. It is not known, of course, in what extent Marity influenced the text of the 1905 relativity article.

But clearly, Grossmann has a strong role in the birth of General Relativity via non-Euclidean geometry to which he specialised soon. Love Letter #48 clearly states that at the end of 1901 Einstein still does not know what is that.

Very few knew the topics at the beginning of XXth century. I guess that Grossmann may have got his starting interest about non-Euclidean geometry in his Hungarian schooling. The end of XIXth century brought again a wave of self-pity and self-condemnation that we have geniuses and neither we, or foreigners honour them until it is too late. Specially, in the 80's and 90's the Hungarian scientific community much discussed the merits of Bolyai Farkas & János, and some of this discussion trickled down to schools/journals.

So it seems that the memory of Bolyai János helped the birth of General Relativity in the above way.

The professional and social successes of the two Bolyais, father and son, much differ. The father, Farkas, may have been not just at the level with his co-student Gauss, one of greatest mathematicians of all ages, elaborator of the Easter Formula, elaborator of a practical and fast method for calculating planetary orbits for asteroid research, elaborator of analytic handling of any curved surface, father of Gaussian distribution &c; and in later years he was much poorer than State Councillor Gauss. Still, Farkas was too a leading mathematician, he had his family lands, and in later years his existence was consolidated. He was a man of Society. He did not make too much politics. Still, he was a fine exemplar of Hungarian Reform Age. He wrote historical plays. With a single exception [14] I do not know about their theatrical performances, but thery were published in a substantial Volume [15], and knowing the Reform Age, I am sure that they were read by "other patriots". He published a lot of mathematical books, mainly for teaching, both in Latin and in Magyar. He invented new types of ovens, both for kitchen purposes and for heating the rooms [16]. The ovens were much better than the old ones, and they were indeed mass produced. He invented a horse-cart in which it was possible to sleep, to heat &c.; something necessary in 1830 Transylvania, with heavy winters on bad intercity routes, still 50 years till dense railway networks. It seems that this idea was not adopted; but people knew about. And so on, and so on.

So when the Magyar Tudós Társaság (Hungarian Scientific Society [G]), in some years renamed as Magyar Tudományos Akadémia (Hungarian Academy of Sciences) started to work, soon he became a correspondent member (see e.g. K 23/21). Even now this is the second best and honoured status for a Hungarian scientist; the first is of course to be full member. The Secretary of the Academy wrote him a letter, and this letter tells that were the Tentamen written in Magyar instead of Latin, B. F. would have been elected a full member. (The first and foremost branch of the Academy is for Magyar linguistics, and she was founded on first place to help the evolution of language.) Bolyai Farkas was honoured in his life; and also when the Agropolis Calvinist Church erected him an imposing tombstone of tall black marble. As an example I can tell that in the time of the strange honeymoon of bólyai Bolyai János and kibédi Orbán Róza Bolyai Farkas earned from the Academy (simply for being correspondent member) cca. 4 times of the salary of housekeeper Orbán Róza, and of course this income was secondary. He was Professor, &c.

Now let us turn to the congenial son. Without doubt he solved first a problem haunting geometers for more than 2000 years. It was a real, well known and disturbing problem. Note that he was impossible to be elected an Academian: he did not have any publication in Magyar.

His priority claims relative to Lobachevskii [H] would need elaborated studies, after 70 years of Soviet Union and 45 years of Soviet dominance in Eastern Central Europe; I make only superficial notes in Appendix B.

However, what about priority with respect of Gauss? Formally no problem. Gauss did not publish about the axiom of parallels until 1832, when he definitely got the Appendix. However there is Gauss' reply suggesting that he had similar results unpublished. Now, I think here a Riemann geometer (as, for example, I and Grossmann, and definitely not Marity or Einstein) can tell something.

First. If a mental giant of Gauss type reads the Appendix, he gets the trick. Even if he did not have the result, he sees how he could have done; and with minimal fighting nature he would not tell "Sir, you discovered something which I did not!". So his mild answer does not prove anything at all.

But, second. Surely he attacked the same problem previously; only from a different angle. He elaborated an analytic formalism: functions describing the surface in different coordinate systems. One should check the timetable but I am sure in 1832 he could have been able to handled a Bolyai surface in 2 dimension and he was able for a spherical surface.

However it is quite a different matter to do it axiomatically and in 3 dimensions. With my Riemann geometric experience now I of course can do anything in a hyperbolic space; but still, with the whole 170 years, it would take for me a lot of time to prove what Bolyai did: that the axioms are complete without that of the parallels and that they are not self-contadictory with the alternative axiom. (While I can construct the ultraparallels without any problem, of course.)

So I am reasonably sure that Bolyai has priority even to Gauss' unpublished papers in Euclid's parallels. The problem was a real problem, much attacked and its solution may have brought great international honour.

It may have; and it did not. Why?

The obvious (for a Hungarian) explanation is not true. Bolyai János did publish the result. He did it in a clumsy way as an Appendix of his father's textbook. The printing office was very, very slow. (Years...) But when it appeared, it was still in time. Also, it was not published in Magyar (another frequent bad habit of ours). It was in Latin. Not a language in fashion anymore on the West, but still intelligible. (Anyway, even now some mathematics appears in French which is a 2000 year evolved form of Latin; and still can be understood.) The PR work was bad; Gauss got it but Gauss' concurrents did not. Still, it seems that Bolyai János' professional relations were almost nonexistent.

He was unknown at University circles. The nearest universities were Pest, Vienna, Krakow and Prague. We can strike out Prague: until 1920 that university was rather exotic for Hungarians. Krakow was not. But the two natural connections would have been Vienna and Pest. Vienna: where the central office of the Imperial-Royal Army also was situated, and Pest, the only university of Hungary. OK, none of them was really strong in mathematics. Vienna was strongest of the world in meteoritics [17] and Pest was good in theology (a lot of religions there) and medical sciences. But they were Universities. It seems Bolyai János did not know any mathematician there. His father did not either; but it happened because of the Protestant grants from Germany, Holland and Scotland for Transylvanians. But Ing-Cpt. Bolyai would have been tolerated at two Catholic Universities, Vienna and Pest...

Now an obvious explanation would be the timidity of some high-breed intellectuals. Indeed, later Orbán Róza can bully the Ret-Cpt. Still, there is a probable story about a duel sequence with his fellow officers, some dozen on one day (with heavy cavalry swords), among which Bolyai János played on violin. If the story is true, János was not timid at all.

This is supported by his (of course unpublished) Curriculum. There first he positively speaks about ancestors performing heroic and/or hopeless acts. Then comes his father [@776]

We know that he is generally not without critics about his father. But now he is positive. In Germany his father behaved himself as fit for a Hungarian nobleman (see Appendix C) among mere burghers and Germans. It seems that the tobacco was forbidden in some German cities. Now, János writes that somewhere Farkas tossed a soldier to the wall because he wanted to take away his pipe. The Passau story is even more vivid: Farkas starts to smoke his pipe "quietly" in a cafeteria. The multitude wants to oust him from the cafeteria; Farkas "did not take the situation as a joke", "with his sword brandishing above his head"..."promised immediately death"...so the German burghers "calmed down and ceased their design". The picture is vivid. The Hungarian mathematician, who is a nobleman, performs a forbidden act in a German cafeteria. The locals want him oust, but he is a Hungarian nobleman (surely with large mustacchio?; long hears are explicitly mentioned), so he menaces them with death, so the mere burghers cease to force local law. So, while János is critical to his father in general, he is not when he defends his rights (including nonexistent rights implicit for a nobleman). So János is not timid at all.

Maybe we get some insight into the problem by a letter of Bolyai Farkas to Gauss in 1816. Farkas wants to send János to Göttingen for 3 years, and of course, his idea is that the boy should live at his friend Gauss. He would pay, of course, but "...tell me, do you have no daughter who could be (reciproce) dangerous for him?... Are you healthy, not poor [for health]?... Is her ladyship your wife, an exception from among her gender...? [9]. Now, the successful Gauss was not a nobleman. To be friends at the University is one thing. To socially mix another. A mesalliance...? And this is Farkas, who is much more friendly than János. János is practically unable to make friends at all. (Or, is this true? It seems so, but nothing is absolutely sure; wait for the Appendices.)

Between 1834 and 1846 János lives in Domáld/Viisoara, with the "housekeeper" Orbán Róza and two small children. When anybody else would take the consequences of his big success and would advertise his greatness, he almost stops. Except for the unsuccessful application with Responsio, a complex algebra paper, to a Leipzig competition we do not know any greater work until Salvation Theory starting in 1841.

Maybe father and son have had quarrels in János' youth. It is not rare to have such quarrels. This must not make a son stop in production.

Maybe the reason not to work in geometry was the lack of speaking with anybody about it. Of course, there would have been his father, but the distance is 40 km on rather bad roads. In addition, the Bolyai family hated Róza [I], see e.g. [@438]. This point deserves some discussion, but only for half of the population, so it goes to Appendix D.

No doubt, Bolyai János was a first-class mathematical genius; otherwise he could not have solved the problem of parallels. (OK, now we know from Gödel's Theorem that in some sense no axiomatics is full. And then what? He did make an axiom system looked for 2000 years.) Also, no doubt, he was very weak in social connections. However even then, two additional facts were necessary to the negative result. First, nobody really honoured him for his great success. Of course we shall never know his father's opinion (it was verbal), but we know Gauss' answer, and we know the opinion of his military superiors. Of course Gauss was envious and the opinion of the military leadership is professionally irrelevant; still, we are humans. If we perform something nice and nobody praise us...

Of course it was mainly his fault that from 1833 he was unable to discuss anything with anybody (except, of course, the housekeeper). Still, nobody seems to be interested in the reactions of the other side. For example, what about the Agropolis College? The son of the department leader is at home, he solved the problem of parallels, still no discussion. Similarly, the Hungarian Academy of Sciences got the Appendix. Of course nobody has to be interested in somebody else's result if the author is obviously not interested; still what about the Pest mathematicians?

All his people (except of course Orbán Róza) practically boycotted the retired Captain. Very probably because of his style, but boycotted. Then the retired Captain responded likewise. The final result is absolutely nothing. Not a unique history in Hungary.

But: Bolyai János is the first human solving the problem of parallels. His soul was reckless; now may his shadow rest in peace. It probably will, having seen that we pay the proper attention to his success.

Hungarian MEC-00736/2002 grant covered my participation on the starting Bolyai conference in Agropolis, July 3-6, 2002.

I have no quarrel with History of Science. However it seems that Bolyai János’ Life and Work is a heavy task; empathy, Riemannian Geometry, Russian linguistics and such are all needed.

Comments are welcome, and I want to refresh the text from time to time. I may be ignorant in a lot of matters discussed here; only I seem to see that I am not the only one.

I have recently learned that historians Kiss Elemér, Oláh-Gál Róbert and Weszely Tibor just started to publish results altering the traditional picture. Maybe in the near future a real synthesis will be possible. Let us hope.

Hungary had the official language Latin bw. 1000 and 1841, and while local languages were used automatically in everyday life, until cca. 1830 they were never used in higher administration, excepting Transylvania. In Transylvania, while the official language remained Latin, Magyar and Saxon gradually appeared even in administration after 1541.

Hungary was divided into counties of changing number but cca. 50, plus sedes. The latter term is Latin, and cannot be translated into English (or anything except Magyar, Slovakian, Saxon & Roumanian) but a sedes is something which is i) not a county; ii) but is also a partially self-governing territory; iii) in some sense it indicates a kind of autonomy; and iv) it does not know the dichotomy of noblemen/commoners. So the territories of Jazyges & Cumanes were organised into sedes outside of Transylvania, and in Transylvania territories of Saxons & Szeklers were sedes. (So in Transylvania county territories were Hungarian (speaking mainly Magyar & Roumanian), sedes territories Szekler & Saxon (speaking Magyar & Saxon/German); but Hungarians, Szeklers and Saxons, the "tria nationes", were all Hungarians in the sense that they were citizens of Hungary.

As for the 3 localities always appearing in this HTML, the biggest, in the Szekler Sedes Maros (Marisus) has the Latin name Agropolis, so this was the Hungarian name. The Szekler name is Székelyvásárhely, the Saxon is Neumarkt. As for modern names, the Magyar/Roumanian is Marosvásárhely/Tirgu Mures. (Note that by etymology Tirgu = [Finnish] Turku [Market Place = Vásárhely], but there in Finnish it is a Swedish loanword, this is the reason of the similarity to Indo-European Roumanian.) Dolmánd is Viisoara in Roumanian; it seems that it never had a permanent Saxon name and the Latin name seems to have been incidental. As for Bólya/Buia the Latin name must have been Bolia; the Saxon is Bell. Dolmánd/Viisoara was in County Küküllô.

If somebody finds this complicated, I cannot help it. The complications motivate it to be gone into an Appendix.

As for demonstration, see the capitols of Hungary. The first capitol, from 1000, was along the Danube, a city called Strigonium in Hungarian, i.e. Latin. In German it is Gran, in Magyar Esztergom (and it has no different Szekler name; it is out of Transylvania). Slovakian linguistic border is not far: in Slovakian the city is Ostrihom. The city of coronations, however, is different: it is Alba Regia (of course), in Magyar Székesfehérvár, in German the improbable Stuhlweissenburg. The Slovakian name is irrelevant, but the Turkish not (I am serious): Ustolni Belgrad. (And do not confuse it with the Transylvanian Alba: Alba Iulia/Gyulafehérvár/Weissenburg/Balgrad.) Then in 1250 the capitol was transferred to Buda/Buda/Ofen/Budim/Budun. (The fourth is Slovakian, the fifth Turkish.) In 1541 the Ottoman Turks occupied Buda/Budun, so the capitol went to the extreme West to Posonium/Pozsony/Pressburg/Presporok; the fourth is the original Slovakian name but Slovakians in the XXth century accepted a Bohemian theory that about 860 a local count Braslav/Vratislav may have had a moderate fortress there called Blaslawspruch by a Bavarian annal. (The existence of Braslawspruch is very probable, the exact place is unclear. Also, the count may have been a subject of Eastern Franks, Great Moravians, Duke of Croatia &c.). The present capitol Budapest is a result of the merging of the twin cities Buda & Pest.

Each Szekler speaks Magyar as first language; in average better than Magyars do; they are Hungarians when compared with anybody outside of Hungary and they are Magyars when they are not compared to a Magyar who is not Szekler. Between 1541 and 1867 the Transylvanian internal law knew about 3 nations: Hungarian, Szekler and Saxon. According to Szekler opinion Szekler sedes existed due to an international agreement of Szeklers and settling Magyars in 896 (Hungary did not yet exist), so the unilateral Hungarian reorganisation in 1872 was seriously problematical. But our time horizon ends with Bolyai János' death in 1860. Also, the Austrian annexation of Transylvania in 1690 (using the almost defunct Empire of Germany as a figurehead) was considered illegal, and Kings of Hungary and Archdukes of Transylvania (both the same person as the Emperor of Germany except the time of King Maria Theresa of Hungary 1740-1780, who, being female, could not be Emperor, of course) started a gradual unification of the chancelleries of Transylvania & Hungary, both in Vienna, according to the demands of the Parliaments in Posonium and Alba Iulia. But when the story starts, Agropolis is still may be visualised (erroneously) even as being in the Empire.

I stop here for your sake although I could continue.

Since 1945 we in Eastern Central Europe were learning a regional History of Science. E.g. radio was discovered not by Marconi but by Popov. Space travel was suggested by Tsiol'kovskii, not Goddard. Conservation of matter was discovered by Lomonosov, not by Lavoisier. Sometimes steam engine was discovered by a Russian peasant, not by Watt. Post-WW2 science history works are not necessarily useful because it was professionally dangerous to refute Soviet claims, while in Hungary nobody took such claims seriously.

The problem is that in Soviet Union a retroactive history was in practice, with mass forgeries by retouching photographs, unreachable newspapers and journals, huge closed sections of libraries &c. I think that some Soviet claims were genuine (e.g. Lomonosov seems to have performed some experiments in sealed containers) but we cannot know which ones. Look at the "New Chronology" of Fomenko in Russia, claiming that all history of the World is forged until 1300 (no ancient Rome, Jesus was born in Antioch in 1054, Moses was his uncle &c.) while in Russia until 1682.

The German publication of Lobachevskii [2] is from 1840; the Appendix is published in 1832. Gauss read it not later than March 6, 1832, and the fact is on Bolyai János' military sheets for Year 1832. Soviet historians mention Lobachevskii's publication in "Kazanskii Vestnik", cca. the "Herald of Kazan", in the Volume 1829, in Russian, and a lecture from 1826 (whose text is unknown). Then the Hungarian side found some evidence about a manuscript of Bolyai János. He is said to have given the manuscript in 1826 to his superior officer and ex-teacher Wolter von Eckwehr in the Arad fortress; he did not get back. The fortress of Arad exists, von Eckwehr did exist, the manuscript is not found but can hide in any archive of the late Imp.-Roy. Army. In addition some Hungarians can argue that Bolyai was ready with the construction in 1824. Let us stop with this.

The problem is that we know very little about Lobachevskii's early works. We know the title of the 1826 lecture: "Voobrazhaemoi geometrii", cca. "Imaginary geometry" [18] which can mean a lot of things, and indeed some historians tell (I do not know the evidences) that Lobachevskii tried to prove the Euclidean postulate of parallels there. As for the 1829 article(s) first it is 1829/30, second, Kazanskii Vestnik became the journal of the University only in 1834, when rector Lobachevskii re-organised it. Third, the title was very general: "O nachalah geometrii", i.e. "On the Beginnings of Geometry", or "On the Fundaments of Geometry" [18] (both translations are possible); Academician Ostogradskii (you may remember the Gauss-Ostrogradskii Theorem about volume integrals of divergences) reported that it contained errors and the Academy should not do anything about it. It is possible that Ostrogradskii was in errors; but somebody should tell definite statements what is in "O nachalah geometrii" published in "Kazanskii Vestnik".

And now some contradictions, without suggesting anything at all, except that they prove that the question has not been studied with sufficient rigour. What was Lobachevskii's published result in Kazanskii Vestnik? This should be the real question behind the priority argumentation. Namely, if Lobachevskii published something which was a viable 3-dimensional non-Euclidean geometry, then he indeed has priority in non-Euclidean geometry (except if somebody sometimes finds Bolyai's Arad manuscript from 1826, e.g. in a Vienna Kriegsarchiv, for which chances are infinitesimal). But if this 1829/30 publication cannot be found, or it contains conjectures, or erroneous constructions or such, then with the Appendix to Tentamen Bolyai has the absolute priority. I note that in any case Bolyai has the priority for absolute geometry, that without the axiom of parallels. That was not found in any of Lobachevskii's papers.

Then, first: where are the 1829/30 articles? The answer is ambiguous. We have the information that it appeared in several parts. Ref. [19] informs us that it appeared in N°'s 25, 27 & 28 of Kazanskii Vestnik. However both Academician Ostrogradskii from 1832 ("...Kniga g-na rektora Lobachevskogo oporochena oshibkoi...") and the angry editorial of the journal "Syn otechestva" (Son of Vaterland) from 1834 (..."Lobachevskii...napisal s kakoi-nibud' ser'eznoi cel'u knigu...) [18] mention "kniga", i.e. a book. But, either a series of journal article or a book: where is it?

For this we have a double answer. 1) It is of course in Kazan [20]; 2) it is unavailable [21]. Answer 1) is given by a Russian Internet site; Answer 2) is that of a US university library. Since Answer 1) does not need too much comment, let us see Answer 2). It tells that: "No copies of Lobachevsky's first publication of his non-Euclidean, hyperbolic, geometry in the Kazan Messenger are known to exist. This French translation [at Gauthier-Villars, 1866] is the earliest publication available". To be definite, these sentences appear in the online catalog of the William Marshall Bullitt Collection of Rare Mathematics and Astronomy, of the Ekstrom Library at University of Louisville.

Now, the Louisville statement may or may not be correct. I am not a librarian. But if it is correct then we have only hearsay about the crucial 1829/30 text! The contradiction between [20] and [21] belongs to librarians; but it is a fact that in all geometry and history papers which I read, the way of exposition is: Lobachevskii constructed the non-Euclidean geometry, see the German paper from 1840 (that is [2] here), but he published this in Russian already in 1829 in Kazanskii Vestnik. Briefly: I could not find an author who would have read Kazanskii Vestnik; and in Louisville they believe that it is impossible anymore.

Now let us see a source which seems to know a lot about Lobachevsky: Vera Fr.: 20 Matemáticos Célebres [22]. He tells that Lobachevskii was the Copernicus of Mathematics; and that he was anti-Kantian. He also tells that for Lobachevskii "...the sum of the angles of a triangle is less than two right angles; in a point there are two parallels to a straight line...". The same as Bolyai János' idea. If they are in the Kazanskii Vestnik articles, they are earlier than Appendix to Tentamen.

However just before this Vera writes: "In the Geometry of Lobatschewski a straight line can be perpendicular to itself..." (in the original Spanish "...una recta puede ser perpendicular a sí misma".

If this is also in "The Fundaments of Geometry, Kazanskii Vestnik N°'s 25, 27 & 28, then Lobachevski’s Geometry is not a correct theory of Space, and then it has no priority to Bolyai. Namely, in a non-Euclidean Space of signature (+++...+) in any dimension with any curvature &c. a non-null vector vⁱ has a positive norm, length, absolute value &c. (g_rsv^rv^s)^1/2 and orthogonality of u and v is g_rsu^rv^s=0, so no vector can be orthogonal to itself, a sí misma. If there is a (++...-) signature, it is possible [23], but then it is not a Space, but a Space-Time. In our space-time the signature is (+++-).

And, look. The "fact" that to a straight line at a point there are more than one parallels (2 or infinite; that is matter of definition) if the sum of angles in a triangle is less than 2 right angles is not a new discovery of Lobachevski (or Bolyai); it is older than Euclid. For simplicity, I cite the opposite statement from Aristotle: Even parallels meet (in finite; or somebody would tell there are no parallels) if the sum of angles of a triangle is greater than 2 right angles [24]. The statement is correct. In spherical geometry the role of straight lines are played by biggest circles; e.g. longitude lines of Earth are such, they are as parallel to each other as possible, still they meet at the two poles. In a hyperspherical 3 dimensional space [7] any two geodesics ("straightest lines") meet. Aristotle knew this and published millenia before Lobachevskii. (The Aristotelian canon doubtless existed in 60 BC, and the first known editor is Andronicus of Rhodes.)

I think any historian interested in the priority argumentation should try to read Kazanskii Vestnik N°'s 25, 27 & 28. I am not a historian.

To be sure, I do not suggest that Lobachevskii would have copied Bolyai in 1840. Tentamen/Appendix was a college textbook. In principle it may have arrived at Kazan University in 8 years, but that would have been a sheer accident. Kazan is along the Volga River, without Transylvanian connections. The book is in Latin, not read at Greek Orthodox Great Russian circles. And, in a minor point, Lobachevskii is less general in 1840 than Bolyai in 1832. But also, it was practically impossible to get Kasanskii Vestnik in Temesvaria (Temesvár, Temeschwar, Timisoara) or in Lemberg (Lviv) just after 1830 (anything was written in it) where Bolyai was then an engineer of the army; and Bolyai did not read Russian. (It is written in Cyrillic.) Most probably up to 1840 Bolyai and Lobachevskii did not know even the existences of each other.

There is a popular poem of our very popular poet Petôfi Sándor from 1847. The poet (2 years later he would die in the freedom fight against overwhelming Austrian & Russian troops; son of Cuman parents who were originally Slovakian) dined with a local nobleman, whose food was good but whom he found too conservative. The famous lines (of course translated here without verse) go as "I do not write, nor read; I am a Hungarian nobleman". Hungarian noblepeople went then up to 10 % in novels and <3 % in today's sociology, which is a problem, is connected somehow with the Szeklers &c., should be solved but will not done now. I only tell that in Hungary a mass of poor nobles existed, just as in Poland & Croatia. The great quantity and political rights gave a self-confidence. So "I am a Hungarian nobleman" was a popular slogan.

Now I give this slogan on 4 languages of Hungary; it practically would have had no meaning for a Saxon or a Szekler. So:

Ego sum nobilis Hungarus (Latin).

Én magyar nemes vagyok (Magyar).

Ja som uhorsky sl'achtic (Slovak).

Eu sim nyemnyis ungur (Roumanian).

The meaning is: I will not pay at bridges; go to hell; I have voting power, &c. But observe that Hungarus ~ uhorsky ~ ungur comes from the name of the country (Hungaria) and finally from the name of Onogurs <- On ogur = 10 tribe [K], a Chuvash Turk alliance in the VIIIth century, while the Magyar form comes from the name of Magyar ethnic. Was then colleague Pisút [G] not absolutely right?

The Superman has his Earthly name Kent Charles, but he is originally Kryptonian. Now, he is Superman, because he is very strong, can levitate, &c. His young sister is the Supergirl. For men, to be strong is Super.

However I doubt that Earthly women would accept somebody a Superwoman simply if she is strong and flies. For different viewpoints, you may read e.g. Niven Larry's Men of Steel, Women of Kleenex.

I must admit that, being a male, I do not know exactly who is a Superwoman. Women have the right to decide. However I can make intelligent guesses.

For an analogy we can use Suda. If we use it, we can demonstrate that we are true scholars. Namely, Suda (or Suida, or Suidas) is a compilation from Xth c. AD, Byzantium, surviving up to now, using much older sources; more or less a lexicon. In this time there is a Suda on Line.

Suda, of course, gives entries for Aristotle, son of Nicomachus, the philosopher. One of the entries gives a surprising information about the origin of his wife Pythias, daughter of Hermeias of Atarneus, Tyrant of Assos, but I plan to write something about Hermeias, so at the present you may be content with [25]. However, there is another information about Pythias, daughter of Aristotle & Pythias.

"The daughter of Aristotle married three times and after giving birth predeceased her father Aristotle" [26], [L]. Why this information?

Of course, Suda was compiled by men. Still, it can be used because they observed an unusual (for ancient Greeks) ability. Look, Pythias, daughter of Aristotle, died in purpureal fever, so probably not older than 40. If she predeceased Aristotle, it happened less than 25 years after Aristotle's arrival to the State of Atarneus, but Suda is not too reliable at some points. Anyway, the younger Pythias married three times in her youth. Most Greek women married once, some twice. She was able to do thrice. (One husband, according to other sources, was a high-ranking Macedonian officer, commander of the Peiraeus occupation troops [27]) That was an outstanding ability for women in ancient Athens!

Now consider Orbán Róza. OK, she got a Captain. Not too bad, but I am sure, some Szekler girls got Majors, maybe one or two even Ltn. Colonels. And anyway, the Captain married her only when he could do it without caution money.

But she was able to borne a son 5 (or 3?) years after divorce whom the father officially accepted! This is something. The details should concern historians of Science; but Róza must have had either outstanding sex appeal or outstanding persuasion power. Of course, she could speak Magyar excellently, since she was Szekler.

Two authors called attention to a letter of a nephew of Bolyai János, Bolyai Gáspár [28]. In his 78th year, 1932, he narrates a story from 1858 (when he was 4 year old). If he remembers correctly, Bolyai János was a jolly fellow. In his story the Bólya landowners and Bolyai János go to Seben/Hermannstadt/Sibiu, to drink Orlát beer, better than that of Hermannstadt; they eat and drink. In the morning the whole company drinks Rostopchin liqueur and they are happily drunken. They go to Bólya, for a big diner, and drink strong vines until next morning. Bolyai Gáspár is a close relative and he may have preserved family traditions. The described lifestyle is diametrically opponent to the Salvation Theory.

An Internet site [29] tells that Bolyai János could have had great effects on ladies: an example is Orbán Róza, the relatively well educated "noblegirl" (this notion is impossible in English, but is a mirror translation of expressions meaningful in Hungarian languages), who eloped with him. Most historians regard Orbán Róza as his paid employee. I do not try to settle this here.

Quite recently a researcher found a fourth possible child, a daughter from 1844. The girl seems to have been named as kibédi Orbán Klára, and had this name until Bolyai's death. From that time she is Bolyai Eliza [30]. Such a drastic renaming seems to contradict XIXth century Hungarian or Szekler laws, but the Bolyai-Orbán family deserves more research. (Is this phenomenon by any chance analogous with Saturnian moons Ianus and Epimetheus, mimicking each other on a common orbit?) The discoverer promises more publication on Eliza. His opinion is that all the children are from Bolyai and when Orbán Róza announced otherwise, she "only would have liked". Also his opinion is that Róza was illiterate.

Kiss Elemér’s new results, got by reading all the 14,000 pieces of the Bolyai material, show that in the Domáld "hermitage" Bolyai made essential research in number theory, never published. Also, his father once suggested him to write another Appendix to another textbook, the unpublished Responsio about complex number theory. However he should have paid the respective part of the printing process.

Kiss’ conclusion is that, provided he wanted to publish, the years after 1834 would not have been without success. Kiss published an English book about this in 2001, but I did not yet see it, so I give other references as [31], [32], [33], [34] & [35].

He also mentions a strange event. In 1844 the Hungarian Academy of Sciences set a smaller prize for studies on the problem (!) of parallels, solved by Bolyai János 13v years ago.

The Tentamen + Appendix was in the library of the Academy (!) for 9 years. Before anybody would think about stupidity, remember that Bolyai Farkas, author of Tentamen and Father of János also was a member of the Academy! My guess is that the Pest mathematicians were as uninterested in the Bolyais than reciproce (to use a favoured word of B. F.).

The Houses of the Bolyai-Orbán Family

In the last 2 years Oláh-Gál R. and Weszely T. somewhat clarified the Agropolis years 1846-60, see [36] & [37]. More than for a century we knew the report of an eyewitness [38] telling that in or just after 1846 B. J. built his own house. The story is that he did not have license, but he first screened the territory. Then somebody appeared from the civil administration: B. J. menaced that fellow with stabbing. Then came the military, to which B. J. told that he would make consecutive duels with the army. Then they went away. See the analogy with the Passau tobacco affair of his father.

Now the researchers disprove the story and can identify the building. The building was the property of Orbán Róza. Also they identified B. J.’s last flat, and somewhat clarified the basic chronology of the Szôts Júlia-romance.

Research definitely speeded up in the last 2 years.

My personal opinion is that the Bolyai-Orbán relations would need at least a Subcommittee, balanced in members between men and women, and Hungarians and Szeklers; otherwise male/female chauvinism and similar influences would keep to distort the picture. Also, note that a picture of Orbán Róza would help the understanding, but even a reliable Bolyai János picture is unavailable.

In the XXth century we learnt that History can be changed retroactively. Mostly we believe that there is ignorance or political manipulation behind, but a small minority even believes in Retro Causality behind. Analogies/homologies are yielded by Quantum Mechanics and General Relativity. We will not discuss the physical aspects here; the other 2 mechanisms are well known.

Maybe the most general attempt to discover Falsified History is that of Academician (differential geometry) and Department Head Fomenko Anatoliy T. in Moscow. Those studies are amply being discussed in recent years, but I can also cite a French author from 1835 [39] founding a theory that Napoleon never existed, is a mythical hero, a version of the antique solar deity Apollon.

Now, it would be better to see how alternative histories are generated. For example, we may assume that the main mechanism is ignorance. If we do not know some details, the incomplete puzzle can be fitted together in different patterns, and then the result is a few different histories.

This surely do happen about very old events about which written documents are scarce or nonexistent. However the life of Bolyai János is a good example for alternative histories from the XIXth century in Europe, with printing, journals, newspapers and modern administration. Since the story may serve as analogy to a lot of other cases, let us see now the sketches of alternative biographies of Bolyai János. I do not want to discuss further the priority; that is mainly a geometrical problem + ignorance.

Born in 1802, Claudiopolis/Kolozsvár/Clausenburg/Cluj/&c. His father imbues him in childhood with the love of Geometry. Because of poverty he goes to the Military Engineer Academy in Vienna and focuses on Mathematics. Then serves at Arad, Temesvar/Temesvár/Temeswar/Timisoara, Lemberg/Ilyvó/Lviv and Olmütz/Olomouc. In 1823, from Temesvaria he sends a letter to his beloved respected mathematician Father that he just have generated a whole World from Nothing; surely he meant Hyperbolic World. In 1826 at Arad he gives a ms. to his superior for advice; it is the complete text of the Absolute Geometry, but the stupid Austrian military is not interested. In 1831 his work is published. However jealous Gauss belittles it. Then he wants to work more on his ideas and asks for furlough for it; but the stupid Austrian military puts him on pension.

He returns home, and quarrels with his father who takes Gauss' side. To be able to work more he goes to Domáld, lives in poverty, but works hard on Mathematics, Salvation Theory (a developed version of the ideas of great Rousseau) and other important ideas.

However he cannot work without perturbations. First, his health is rather poor; he suffers from many illnesses but heroically ignores them when working. Second, his housekeeper always disturbs him. She quarrels, grudges, and, by some plot, bears him 2 children: Dénes & Amália. János is a good father of these children.

In 1846 his father ousts János from the Domáld farm. End of peaceful life. In 1848-49 János is for the Hungarian Revolution & Freedom Fighting; he offers his service for Governor Kossuth but ignored. He recognises his duty for the children and marries the housekeeper. However Austrian reactionary military wins and his wife remains impossible. In 1852 János leaves everything to his wife and the 2 children and goes away. In his last years he lives a small flat in serious poverty, with a single maidservant & nurse, Szôts Júlia, amongst lot of ilnesses. Died 1860. 01. 27. Buried on 29th, with only 3 mourners + military. Afterwards the Austrian military temporarily seizes his manuscripts.

Resurrection starts in 1867, when, in the year of Hungarian independence, the Bordeaux Academy republish the Science of Space.

This has been written in the spirit of Romantic School of History of Science. The reader may remember books or movies about Évariste Galois, or Giordano Bruno, or Galileo. And this text is essentially the Hungarian majority opinion since 1870. However the above story contradicts several documents and most definitely, its spirit contradicts the last letter of nephew Bolyai Gáspár, last contemporary of János. So let us see a diametrically opponent biography, supported in spirit by Bolyai Gáspár, but in some details by a lot of historians, and by some other documents. All the used documents have been cited above and many summarized too.

Born in 1802, Claudiopolis/Kolozsvár/Clausenburg/Cluj/&c. Goes to the Army, because of natural inclination. In the 20's his life is vivid, he collects lot of ears in duels and one sequence of duels is famous: he fights 13 duels on a day and as interludes, fiddles.

In the early 30's his health is attacked by a honourable head injury. As an army officer he is crossing from Galicia to Moravia, and Austrian finance people want to check his baggage. It is his right not to be checked, but they are many, so the brutes wound him after a valiant battle.

He elaborates, in his scarce free time, Science of Space, as first such of World. He gets the idea that maybe he could get 3 years furlough referring this. It is unsuccessful but he is able to go on permanent half-pay. In later years, when warring factions try to reactivate him, he is successful to avoid military service forever.

He goes home, settles down in the country house of his father, and elopes with the daughter of the most honoured Kóródszentmárton family. The lovers remain together in spite of many outside plots, and 5 children are born: Dénes (1837), Amália (1840), Klára (1844), Eliza (?) & Gyula (1855). In 1846 civil and military authorities together try to disturb him in building a house to his family; but he is very brave, menaces them with death, and succeeds. He tries to get appointment both from Hungarian and Austrian governments; these attempts are unsuccessful. In 1849, when it is possible, he marries freely his lover. Austrian military threatens the family because of this; then they outwit the Austrians with a fake divorce.

In 1856 he comes to his own, inherits the family lands. Then he lives as fit to country noblemen, eating, drinking & wenching. He believes to be syphilitic. In the very cold January of 1860 he dies in pneumonia, heroically, without a word of pain.

This story is as coherent a reconstruction as the previous one, and is supported by many documents. The only reason that I do not believe in it is that in this scheme the Salvation Theory is mere diversion: Of course for getting a Government job it is good if he is a scholar and in this Scheme he writes a theory for mere reference (since his life is contrary to Salvation Theory, the detailed report of Bolyai Gáspár on drinking, eating & wenching). He writes it along the lines of Rousseau, but amended. So far, so good. But Salvation Theory is some 10,000 page long. Simply to pose with, one-tenth would be sufficient.

The self-inconsistencies in Scheme A and Scheme B are serious. Either you eliminate one half of the story by some historians' method, or:

You take Scheme A) until 1856. Then: the death of his loved and respected Father confronts him with the problems of Life, Afterlife & such. He is stunned by the loss and tells: everything I believed up to now is meaningless. Hence Scheme B:

Born in 1802, Claudiopolis/Kolozsvár/Clausenburg/Cluj/&c. His father imbues him in childhood with the love of Geometry. Because of poverty he goes to the Military Engineer Academy in Vienna and focuses on Mathematics. Then serves at Arad, Temesvaria/Temesvár/Temeswar/Timisoara, Lemberg/Ilyvó/Lviv and Olmütz/Olomouc. In 1823, from Temesvaria he sends a letter to his beloved respected mathematician Father that he just have generated a whole World from Nothing; surely he meant Hyperbolic World. In 1826 at Arad he gives a ms. to his superior for advice; it is the complete text of the Absolute Geometry, but the stupid Austrian military is not interested. In 1831 his work is published. However jealous Gauss belittles it. Then he wants to work more on his ideas and asks for furlough for it; but the stupid Austrian military puts him on pension.

He returns home, and quarrels with his father who takes Gauss' side. To be able to work more he goes to Domáld, lives in poverty, but works hard on Mathematics, Salvation Theory (a developed version of the ideas of great Rousseau) and other important ideas.

However he cannot work without perturbations. First, his health is rather poor; he suffers from many illnesses but heroically ignores them when working. Second, his housekeeper always disturbs him. She quarrels, grudges, and, by some plot, bears him 2 children: Dénes & Amália. János is a good father of these children.

In 1846 his father ousts János from the Domáld farm. End of peaceful life. In 1848-49 János is for the Hungarian Revolution & Freedom Fighting; he offers his service for Governor Kossuth but ignored. He recognises his duty for the children and marries the housekeeper. However Austrian reactionary military wins and his wife remains impossible. In 1852 János leaves everything to his wife and the 2 children and goes away.

His beloved and respected father dies in 1856. B. J. stands at the deathbed and recognizes the futility of Life. He turns to Epicurean teachings. His turnover is operative, since he had inherited the paternal goods, including lands. Then he lives as fit to country noblemen, eating, drinking & wenching. He believes to be syphilitic. He takes a much younger housekeeper than Orbán Róza. Life would be nice, but in the very cold January of 1860 he dies in pneumonia, heroically, without a word of pain.

Of course, this is intended to be a parody. However note that Scheme C explains more facts than Scheme A or Scheme B. The only problem is that Schemes A & B cannot be smoothly joined in 1856. For example, according to History János offers Salvation Theory to Emperor Francis Joseph in 1852 (no Hungarian needs comments here), while in 1858, in the heydays of his gentry life (according to nephew Bolyai Gáspár) János lives very poorly in Agropolis (according to History).

And this is not from Dark Ages of History. The apotheosis of Bolyai János starts in 1867. His children (excepting Klára?) live. I do not know about his wife; but his last maidservant (and lover?) Júlia is available. His nephew Gáspár will live 65 more years.

Of course, there would be a psychoanalytic explanation as below.

According to this "explanation" B. J. would have had two distinct identities, A & B, so Schemes A and B alternate as many times as needed. (Homework.)

However we would like to write History of Science, not a novel.

Bolyai János solved a problem which had been open for 2000 years. He created a World from Nothing. This can be told about only a handful of discoverers, We have to be proud about him and we are, indeeed, from 1867. Nobody was interested between 1867 & 1932 in a true, detailed and self-consistent biography [M]?

[A] There are alternative spellings Bolyai/Bólyai even in Hungary. The reason is that the eponymous village, old estate of the family has both spellings. It seems that the official Nomenclatura from the year of the last pre-World War I population counting 1910 preferred Bólya, but this does not have retroactive force on names. At the end of XIXth century official sources consequently use the name form "Bolyai" so I use it too. Also, for foreigners. Bólya is in Transylvania, but in the last millenium county borders fluctuated. Originally Bólya belonged to Comitatus Alba (County Fehér). Later the county was split into two and Bólya went to Comitatus Alba Superior (County Felsô Fehér, Upper White). However note that Bólya was almost surrounded by Saxon sedes (see also Appendix A). After the 1872 reorganisation Bólya went to County Nagy-Küküllô, but that time is out of the time horizon of the present work. If somebody is curious and has only non-Hungarian maps, the village is Buia in Roumanian, Bell in Saxon. Note that the reflex of Magyar "ly" is regularly "i" in Roumanian, but rather "l" in German. In present Magyar pronunciation the pronunciation of "ly" is [y], written in Magyar as "j", excepting some Western localities where it is "l" and a small area near to Slovakian speakers where the primordial pronunciation "l'" is retained. In Transylvania the Magyar pronunciation of ly is j=[y] for times immemorial; this is shown by the Roumanian reflexes (e.g. Mihály = Michael = Mihai) and the last servant girl of Bolyai János, Szôts Júlia, who announces the death in a letter writes consequently "Bojai".

[B] We arrive again to a constitutional point. In short: the Hapsburg Dynasty, during the anti-Ottoman War of the Holy League (Pope Innocent, Doge Gustinian Marcantonio of Venice, King (Sobieski) Jan of Poland and Emperor Leopold I) got strong positions in the rather anti-Imperial Transylvania and agreed in collaboration with the Prince, Apaffy Mihály I. In 1690 he was followed by his son Apaffy Mihály II. However the Viennese got an idea not to keep the agreement. Leopold I then turned to his alternative title as King of Hungary. In this position (but only in this) he was overlord of the Prince, whom then he transferred to Vienna. Sometimes after 1711 the Emperor gave himself the title Great Prince of Transylvania (only I am afraid that such a title is foreign for English) and tried to rule Transylvania directly (I mean, not through Pressburg). However the constitutional problems became explicit with his successor Charles III/VI.

Emperor Charles VI was Charles III, King of Hungary, so it was very easy to refute his unconstitutional claims in the Transylvanian Parliament without any danger, simply using the number III. (Note that Elisabeth II of England is Elisabeth I of Scotland.) Then, during his successor the Hapsburgs lost the Empire for a time: between 1742 and 1745 the Emperor was Charles VII, the Bavarian Elector-Prince, in war with Vienna. From 1745 the Emperor was Francis I of Lorraine from Vienna, while the King of Hungary was his wife Maria Theresa (she was crowned King, not Queen, and she was Rex Hungariae). Now, who governed Transylvania?

Maria Theresa did, as (Great) Prince, so she had to recognise that the only way to do this is to declare that the overlord is indeed the Hungarian King, not the Emperor. So much that she was titled Prince in 1741, while his husband, as an afterthought, was invested Vice-Prince in 1742, and it remained so even in 1745, when the Vice-Prince became Emperor. From this time a slow and gradual unification started, but still the Vienna Court tried to retain the power in Transylvania. The central role of Vienna was emphasised by the name of the common army as Imperial-Royal, in German kaiserlich-königlich or k-k. Hungarians told that this is impossible and the name should be Imperial and Royal, in German k. und k. because the Emperor and King are not identical persons. This reorganisation, however, did not happen until 1867. Then the army became k. und k., while e.g. the Viennese Ministry of Interior remained k.-k. (working only in the countries of the Emperor of Austria and the King of Bohemia).

So Bolyai János was a Captain of the k.-k. Army. That army was first and foremost Imperial, but Hungarians generally felt it also their own.

[C] Maros is the Magyar name of the big river whence the sedes got its name. The Roumanian is Mures. The Latin is Marisus. All come at least from Scythian or Agathyrsos (see Herodotus, and Aristotle, Problemata XIX.28). Names of big rivers are inherited for long times. By archaeologists interested in Neolithic it is called The Big Eastern River.

[D] Some sources tell that the Bolyai family was of Szekler origin. I doubt this; but independently of remote origins going as far as possible, the family seems to behave itself as Hungarians. Bólya is in the Lakság, they are Hungarian nobles, and the name is Hungarian: the family name comes from the original estate. (Compared to the Orbáns, for example: their family name is originally a given name. They are kibédi Orbáns, while B. J. is bólyai Bolyai.) A Scottish reader may see some points here.

[E] Küküllô is the Magyar/Szekler name of the river. Roumanian is Tirnava. However, as I told just above, rivers keep the name, independently of new people. This is just a good example. Originally Küküllô/Tirnava is Bulgarian, only the first Bulgarian Turk, the second Bulgarian Slav. In Slavic Slovakian "trnka" means "blackthorn", but the berry of blackthorn, "sloe", is dark blue. Now, in all Turkish languages "blue" is "kök/kük"; in Magyar/Szekler it is "kék". In Magyar/Szekler "blackthorn/sloe" is "kökény". So what happened? Somewhere along the river extensive fields of blackthorn were found. Then Bulgars called it "Blackthorn River". In Slavic Bulgarian "Tirnava (Reka)", in Turkish Bulgarian cca. "Kük(üllô)". Then Szeklers took the Turkish variant, Roumanians the Slavic. There is a river "Cerna Voda" in Valachia, Roumania, which is "Black Water" in Bulgarian Slavic, there is another in Transylvania "Karassó", which is really "Kara Su", "Black Water" in Bulgarian Turkish.

[F] My colleague Martinás Katalin tentatively suggested her participation in the project some years ago. Such a team would be extremely balanced and as similar to the Einsteins as possible. One male, one female. One in relativity, one in thermodynamics. One Southern Slav, one not. And so on. But just now she is uninterested.

[G] Magyar Scientific Society is an alternative translation. But Magyars generally translate Magyar as Hungarian in state organisations &c., and the Society, later Academy was founded by the Parliament of Hungary in Pressburg (so without Transylvanian delegates!) with a written Royal consent read in the Parliament. Slovakians tell that the Magyar language is ambiguous in this point, e.g. in 1992 particle physicist Pisút Ján, then Minister of Education of Slovakia, suggested a reform of Magyar language in a political speech; but Magyars in Slovakia refused. Slovaks should call the Budapest Academy Mad'ar, not Uhorsky; however in English text they accept Hungarian. So we can remain with this ambiguity. See also Appendix C.

[H] Lobachevskii wrote his own name as Lobatschewski then publishing in Latin letters. In fact this is the transcription to German. My spelling here is transcribed from the original Cyrillic to English. The same to Hungarian would be Lobacsevszkij. It is in fact transcription to Magyar and to Slovak it would be Lobacevskij (with something on the "c" hard in HTML) and maybe the ending would turn from the correct "-skij" to the "more Slovakian" "-sky". But in English text I must call the official transcription of the Hungarian Academy of Sciences Hungarian and not Magyar. It is not trivial to be Hungarian (either Magyar or Slovak).

[I] Of course, the Ret. Cpt. hated Róza too, many notes reflect this. However it is not the same. Bolyai Antal suggests that the Captain should give some dowry (definitely: zulág=Zulage), then a sergeant would marry Róza and János can get a cheaper housekeeper. What a nice solution, and the sergeant will raise up the children of the Captain as well. The great advance would be the 1 Rhine florin per month saved. 12 florins per annum. Is it worthwhile?

Maybe the psychological key to the understanding of the confuse relation of János and Róza will be the last child, Gyula. (I admit, I cannot evaluate the situation.) János and Róza seem to agree in 1852 that János goes away, leaves 500 Rhine florins for the raising up the children and they do not know each other anymore. Still, when a second son Gyula is born in 1857, after the death of Farkas, János accepts the son. Any comment?

[K] No plural after a definite numeral in rational languages (Magyar, Finn, Chuvash, Turk, Japanese &c.).

[L] Diogenes Laertius knows otherwise. And then what? He is as unreliable as Suda.

[M] Of course, ethics was different in 1867. For 15 years Bolyai János & Orbán Róza lived together without formal marriage. Shocking. It might be denied but the children; then we find as few children as possible. And maybe it was more romantic to look at the Martyr of Science & of Austria than at a real scientist. But it will be a heroic task to clean up the contradictions 142 years after his death.

[1] Bolyai J.: Appendix Scientiam Spatii... in: Bolyai F. Tentamen. Maros Vásárhely, 1832-3. (For a French reprint see Bolyai J.: Mém. Soc. Sci. Bordeaux 5, 207 1867))

[2] Lobatschewski N. I.: Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin, 1840.

[3] Van den Akker & al.: J. Opt. Soc. Amer. 37, 363 (1947)

[4] Wasserman G. S.: Color Vision. J. Wiley & Sons, New York, 1978

[5] Weinberg J. W.: Gen. Rel. Grav. 7, 135 (1976)

[6] Lukács B.: Acta Phys. Polonica B19, 243 (1988)

[7] Robertson H. P. & Noonan T. W.: Relativity and Cosmology. Saunders, 1969

[8] Lukács B.: Acta Phys. Hung. to appear

[9] Benkô S.: Apa és fiú. Magvetô, Budapest, 1978.

[10] Stäckel P.: Bolyai Farkas és Bolyai János geometriai vizsgálatai. Budapest, 1914.

[11] Fráter Jánosné: A Bolyai-gyûjtemény (K 22 - K 30). Bibliotheca Academiae Scientiarum Hungaricae, Budapest, 1968.

[12] Deschamps Dom: Le Vrai Systeme ou le mot de l'énigme métaphysique et morale. J. Thomas & F. Venturi, Geneve, 1963

[13] Renn J. & Schulmann R. (eds.): Albert Einstein, Mileva Maric: The Love Letters. Princeton Univ. Press, Princeton, 1992

[14] Bolyai Farkas: A Páris'i per. Egy érzékeny játék, öt fel-vonásokban. (The Lawsuit in Paris. A sensitive play in 5 acts.) Marosvásárhely, 1818. This play was staged on Apr. 30, 1938, see Ms. 929/e in Manuscriptorium HAS.

[15] Egy hazafi (A patriot) aka Bolyai Farkas: Öt szomoru játék (5 tragedies), Szeben, 1817.

[16] Horváth Farkas: A szobafûtés elmélete. Budapest, 1875.

[17] Lukács B.: Sphaerula 2, 9 (1998-2001)

[18] ***: http://translit-www.klax.tula.ru/~volodin/lobachevsky.html

[19] ***: www.kanazawa-it.ac.jp/news/1999/tech_news1999/dawn990401.html

[20] ***: http://www.gazeta.ksu.ru/10.htm

[21] http://www.louisville.edu/library/ekstrom/special/bullitt/bullitt.html

[22] Vera Fr.: 20 Matemáticos Celébres. Los libros de mirasol, Buenos Aires, 1961. On Internet: wysiwyg://52/http://www.geocities.com/veintematematicoscelebres/cap09.html

[23] Eisenhardt L. P.: Riemannian Geometry. Princeton University Press, Princeton, 1950

[24] Aristotle: Prior Analytics, Book 2 Chaps. 16-19, Bekker N° 68-70

[25] Lukács B.: http://www.rmki.kfki/~lukacs/atarn.htm

[26] Roth Catherine: at www.stoa.org/sol-bin, at search: Aristoteles, alpha,3929

[27] Swiderkowna Anna: Hellenika. Panstwowy Instytut Wydawnyczy, Warsaw, 1974.

[28] Kiss E. & Oláh Gál R.: Term. Vil. 2001/11

[29] ***: http://www.iif.hu/~visontay/ponticulus/hidveres/nemeth-bolyaiak-muh3.html

[30] Oláh-Gál R.: Maros Megyei Népújság, 2001, July 18

[31] Kiss E.: Mathematica Pannonica 1995/237

[32] Kiss E.: Mathematica Pannonica 1997/293

[33] Kiss E.: Természet Világa 1994/405

[34] Kiss E.: Természet Világa 1996/344

[35] Kiss E.: http://www.iif.hu/~visontay/ponticulus/typotex/bolyai.html

[36] Oláh-Gál R.: Maros Megyei Népújság, 2001. 02. 08

[37] Weszely T.: Maros Megyei Népújság, 2001. 02. 13.

[38] Bedôházi J.: A két Bolyai élete és munkássága. Marosvásárhely, 1897.

[39] Pérès, Abbé: Comme quoi Napoléon n'a jamais existé ou grand erratum... Agen, 1835

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