ON BENEFITS OF THERMODYNAMICS

ON BENEFITS OF THERMODYNAMICS

B. Lukács

President of the Matter Evolution Subcommittee of HAS

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

ABSTRACT

Benjamin Thompson, Count of Rumford, was one of the greatest, and surely, most colourful, characters of Thermodynamics. Without claiming completeness, I would like to commemorate him, hopefully also in a colourful way.

1. INTRODUCTION

I am no historian, so maybe the references will be sloppy sometimes. In the future I want to improve the quality of references, but that may wait; real historians can easily find proper references for e.g. the British-American Colonial War or French Ancien Régime. Also, an important part of my information comes from personal communication, and its source may later append this work. Anyways, regard the present study as a First Draft, a Grundrisse.

I am still looking for some original publications of Rumford; in the present study some sources are secondary, and 2 are of unusual forms, a newspaper article [1] and a personal communication [2].

We should have celebrated the quarter millennium of Rumford’s birth two years ago. However I got the information of [1], which is a daily newspaper, just now and I could not wait either until 2014. I promise to update the present study when new stories come forward.

2. THE BEGINNING

Benjamin Thompson (1753-1814) in his youth lived in Bosten, Massachusetts, and then in New Hampshire, near the city Rumford, now Concord. In 1772 he seduced & married a rich widow, so he entered High Society and became a British military officer. Then came the War of Independence. He had to choose between America and Britain, and he chose Britain.

He was not alone at all. James Thurslow Adams, scion of John Adams, the second president of USA, writes that maybe 1/3 of the population of the 13 colonies were loyalists, so wanted the continuation of British rule, and some 100,000 emigrated or rather were ousted [3]. The majority of the emigrants went to Canada; Thompson went directly to London, where he became a deputy minister, then Fellow of the Royal Society, and a knight [1]. As for the Royal Society, we will see, why.

Then he went to Munich, Bavaria, where he became Advisor to the Elector, Charles Theodore, and Head of the Arsenal. The Elector asked the Emperor to make him count; and the obvious choice of title was Count of Rumford. From this point I will mention B. Thompson as Rumford.

Of course, his immigration to Bavaria was not an emigration from Britain. He organised the Royal Institute in London (in 1800) [4], directed first by Davy. Also, his seminal papers appeared in London. (I am still looking for some. My secondary sources [4], [5], [6] mention one from 1789 in Phil. Trans., and another from 1798. No doubt, there were many.) In the forthcoming Napoleonic Era Bavaria sometimes was an ally, sometimes an enemy to Britain, so complications sometimes arose.

3. RUMFORD'S SOUP

Rumford's Soup is remembered even now, and has descendants in cookery. In fact, it entered even the Communist canon, because Charles Marx refers one of its higher quality variants in Capital as a bad example [7]. But really it was better than lots of attempts reforming cookery.

Ref. [1] gives the recipe of ground Rumford's Soup. Rumford was Advisor to the Elector, and Expert in Thermodynamics. The State was pestered by beggars, pickpockets & such, and the idea was to organise some workhouses for them. However on low-calory food ex-beggars & -pickpockets cannot work, even if under State supervision. (Low-calory is an expression from almost a century later. However research towards it started just by Rumford.) Then what to do? This seems a vicious circle: maybe regular work would domesticate criminals, but cheap and still substantial food is to be procured first by State.

But Rumford is not Advisor and Count for nothing. He invents Rumford's Soup as follows. 1 part pearl-barley, 1 part dried (yellow) peas, 4 parts potato, salt according to need, and old, sour beer. You slowly boil it, until it becomes dense. Then you eat it with bread.

Maybe it sounds horrible first. However look. A similar dish (without potato; we are not in Germany) was usual for conscript soldiers in Hungary of the 60’s. Really dry yellow peas are not bad at all if you cook them long enough; of course Hungarian soldiers got it with onions, and instead of barley some wheat flour in it (I would not go into the technique of thickening), and generally got some pork meat chips on it. They did not like yellow peas; they did not like to be conscripted.

Now let us approach the problem theoretically. See a Table for modern composition & calory data:

Raw Food	Protein, %	Fat, %	Carbohydrate, %	Calory/100 g
Pork meat	15	21	0.2	260
Chicken meat	17	4	0	107
Rice	6.9	0.5	79.4	358
Rye bread	5.8	0.8	53.3	250
Dry beans	23.7	2	56	346
Mushroom	1.2	0.2	3.2	20
Potato	1.5	0.2	15.7	72
Pearl barley	10.8	1	73	353
Yellow dried peas	26	2	57	359

So yellow dried peas contain more protein than fresh meat; practically yellow dried peas mimic dried beans, a substantial food. The daily calory need of not too hard physical work is told to be 3400 calories; calculating half of it from bread, the daily Rumford's Soup/person then needed one-third pound of barley & peas, and slightly more than a pound of potato. It is an economic food, and really not extremely repulsive. Of course, vitamin content is another matter. But at least Vitamin B can quite be found in the dried peas, and some Vitamin K in the potato.

Charles Marx criticised the Quality Rumford's Soup, containing also corn and herrings. Herrings contain some Vitamin D. Obviously Rumford's Soup is not the peak of cookery; but it is a good invention for cheap substantial food. Bavarian cookery still produces far descendants of Rumford's Soup, but, of course, with fresh vegetables; Vitamin C is needed.

Rumford fulfilled his duty as Advisor to the Elector, and also his results wee forerunners of the Thermodynamics of Human Body. I think he got the recipe purely empirically, of course.

4. RUMFORD'S COOKING RANGES

The cooking range was amongst the inventions of biggest impact in everyday life. In the times of big families and housewives cooking was a chore. With the invention it remained a chore, but much easier and cheaper. The idea is easy enough, but needs a methodical thermodynamic mind. There are 3 main points of the invention.

The thick iron top of the fireplace permits a continuous change of thermal flux. When you want only keep the food warm, you simply remove it from the center to the periphery.

If you sink some cooking-places well into the iron plate, the pot will have heat also horizontally. First, more heat flux goes into the pot, and less heats simply the kitchen; second, even in a deep pot the cooking will happen more or less homogeneously.

If you have an oven with adjustable flux, you can bake bread, but also can fry anything & everything.

After 1789, abolition of feudal prerogatives (and later by partial massacre of nobles) French burgeoisy started to found restaurants. Restaurants needed fireplaces which were almost as good as huge feudal ones, but were more versatile. (After all, the cook of a Count cooked one series of food on one occasion, while a restaurant has to produce at least a dozen.) Cooking-ranges were excellent, invented just in time [1].

Here I would note a Hungarian scientist inventing various fireplaces not much later. He was Farkas Bolyai, father of János Bolyai, first inventor of Lobatschewsky geometry (about this point see App. A.) He did not publish them, nor patented them, but there was still one such fireplace in use at Marosvásárhely (Neumarkt), Maros County in the time of the 2002 Bolyai Conference. You can consult with [8]; of course first you should learn Magyar.

Then cooking utensils belonged to Applied Thermodynamics. Rumford's Cooking-range was a better and more versatile construction than my gas fireplace which does not contain sunk-in places, so has a lot of heat loss. Also, were I to use vegetable oil instead lard, frying would be really difficult. (Pig lard circumgoes most problems in frying pans.)

5. RUMFORD AND GUNSMITHY

How to manufacture a gun (cannon) at the end of XVIIIth century? Well, first you cast the crude form; from special bronze or brass for ships, and from iron for land artillery & fortresses. Then you bore holes; at least two: the main one and one small for ignition. Rumford, Director of the Bavarian Arsenal, directed the manufacturing of land cannons.

Of course, the would-be cannon is heated up when bored. This was experienced over centuries. But Rumford was interested in Thermodynamics.

Now, look. In that time the leading theory of Thermodynamics is that

Heat Is a Conserved Substance.

If I want to tell two big names, then the supporters are Black & Lavoisier. The next Chapter will be about Lavoisier, so here I concentrate on Black. Black was the teacher of James Watt, who is now considered the "Inventor of Steam Machine", but primitive steam machines worked already for long time in mines. Watt seriously improved them.

However, Black was also the teacher of John Robison, and Robison published a book about Black's lectures [9]. Black was the first methodically distinguishing Heat & Temperature (in later Thermodynamics the first would be an Extensive, but it does not exist, while the second is an Intensive; see e.g. [10], although I should rather cite classical literature. Black's opinion is that Heat is an elastic fluid with inherent repulsion, while "ponderable" matter attracts it. So in empty space Heat is uniformly distributed; bodies concentrate Heat in themselves ("heat capacity"), some less, some more.

Now comes Rumford, supervisor of Bavarian gunsmiths. He observes that friction during boring heats up the iron; and then this heat can be conducted away [11]. Good, Black would tell that friction diminishes the heat capacity, so now the same Heat causes higher Temperature. But Rumford experiences that this higher temperature can be maintained by boring (until the whole hole is not ready; then boring stops for practical reasons). If you bore for longer time, more Heat is conducted away.

But then there is no conserved Heat. Also, the quantity of Heat is proportional with the mechanical work in the process [6]. Then, clearly, Heat is produced by friction, so Heat is a kind of Motion. Also, he tried to measure the weight of Heat Fluid and it was less than 10^-6 part of that of the "ponderable matter".

The matter was not closed: heat radiation was hard to be interpreted in the Kinetic Theory. Still, Rumford had good arguments, and now we know he was right.

6. LAVOISIER VS. RUMFORD: ON HEAT, LADY LAVOISIER OR BOTH?

Now I am in an awkward situation. Some part of the information in this Chapter comes from an external source, which I must cite; that is [2]. But I cannot prove that indeed I tell the original information; and then I would be in trouble. So I tell that I indeed received information about Rumford sas indicated in the References; but you does not have to accept that I received just the information written down in this Chapter. Maybe I do not remember correctly; or maybe I am lying. But maybe not.

I start with a story. In 1987 some problem arose between me and my multiple coauthor Dr. Martinás. It was of multiple origin, but the main part was about the fate of a common paper. Details are not important; this problem was solved in 1998, ten years after the story coming. In an early phase I wrote a letter to initiate the negotiations, and for my mild surprise I became invited into Dr. Martinás' office.

Then I became really surprised. Dr. Martinás waited for me together with a support, the earlier fiancé (from 1970, 18 years earlier). Second surprise was that Dr. Martinás wanted to discuss the Lavoisier-Rumford affair, not our business. The ex-fiancé did not know anything about the Lavoisier-Rumford argumentation, so practically did not participate. I was confronted with the facts first time, but was able to comment the statements, because I had read the book of an excellent Hungarian literary scholar (killed at the end of WWII, while on forced labour) about the Great Necklace Case of Ancien Régime France [12]. I did not check the story coming here; I confide in the History of Science of Dr. Martinás. Anyways, we have many common articles.

I was told that Rumford had an extended visit to Lavoisier in Paris. Then there were high argumentations about Heat Fluid vs. Kinetic Theory. Also that Rumford tirelessly courted Lady Lavoisier, while in Paris. Then, after 1794, the execution of Lavoisier (as a Count, not as a chemist), Rumford reappeared and married the widow. I will continue that Love Story soon, but now let us see the idea of my informer. Surely the two males fought for the female and this was the reason to occupy opposite viewpoints about Heat.

Then came I, arguing according to Ref. [12]. For Ancient Régime noblemen, and especially for Barons, Counts, Dukes & Princes, was definitely not chic to express jealousy, not even to feel it. Ref. [12] collect stories, mostly citing Chamfort. Until I do not find the anecdotes of Chamfort (surely familiar for historians) I simply believe Ref. [12].

One story is about a Count going home. He enters a room, and finds there his wife with "a nobleman" in "unambiguous situation". "By Jove, Madame", tells the Count, "you should be more careful; the door was not closed. Imagine if somebody else entered!".

Another story demonstrates another angle of the problem. Monsieur de Nesle, Madame de Nesle and Duke Soubise are chatting. The Madame is the lover of the Duke. Then M. de Nesle states that "Madame, I heard that now you already have a love affair even with your hair-dresser; this is improper.” Then M. de Nesle exits attentively, making possible for Duke Soubise to slap Madame de Nesle's face. I remember a third story (maybe in a book about Marie Antoinette; I am still looking for it) where the landowner finds his wife in bed with the neighbour landowner. Then he tells: "Monsieur; for me it is duty. But for you...".

The pattern is clear. The aristocracy is not jealous. If the ladies have (open) love affairs with commoners, however, that is shame for the husbands.

So, told I, the construction for explaining the Lavoisier-Rumford argumentation is clearly wrong. Both are Counts. They must not be jealous. Maybe they had some other problem; or maybe they had different views on purely scientific grounds.

The discussion continued. Lavoisier is executed; the widow becomes Lady Rumford. And then the lovers start to quarrel. Count Rumford finds a complicated revenge. The Countess has some favourite potted flowers. Rumford deduces that they would slowly wither away if he waters them with tepid water. He does it; so the Countess is sad for a long time.

Interesting story; Rumford must indeed have been a thermodynamic genius.

As for the specific Lukács-Martinás argumentation, at this point Martinás and the ex-fiancé exeunt; the problem is prolonged for ten years.

7. ARISTOTLE'S & RUMFORD'S LEGACIES

Thermodynamics is the last Aristotelian science. Its fundamental equations are of first order, so there is no Motion without Force. This was succinctly demonstrated by Dr. Martinás, whom, however, is not my duty to advertise. Newton changed Mechanics into second order evolution equations; later other disciplines of Physics followed the example, but not Thermodynamics. That is still First Order; therefore there are serious problems with Extremum Principles, which is one amongst the reasons that we do not know exactly what is minimal in stationary open systems. Extropy does not help because extropy cannot exist; see App. B.

However assume that, by sheer accident, somebody finds the extremum principle. Then Aristotle's 4 Causes [13] reappear. Namely, Aristotle distinguishes 4 kinds of Causes: Matter, Form, Mover & "For the Sake of Which". Now, for our modern reflexes the first 3 can more or less be called Cause; but the 4th is Purpose. Aristotle was grammatically right because both Cause and Purpose answers Why?; but we already know better. Causality & such.

However assume for the sake of discussion that all conduction coefficients are constant. Then in our open system the changes are governed by Minimum of Entropy Production. Then processes go into that direction.

In [14] I discussed some examples. E.g. Planet Venus has a low enough entropy production, because the majority of solar radiation does not enter even the atmosphere; the clouds simply reflect it. Also, during evolution, comparable mammals developed smaller entropy production than reptiles, since their body temperature is higher, so dQ/T can be smaller (the fur is good insulator, so higher T does not necessarily mean higher dQ).

Both processes are Evolution. Now, if Minimum Entropy Production is the Governing Principle, then in thermodynamic context you are entitled to argue backward: Evolution goes to the direction in which it really went for getting Minimum Entropy Production. That For answers Why. Strange, is it not?

Rumford himself taught us two things. First: if somebody learns Thermodynamics well enough, nothing is impossible. With one hand he invented the versatile fireplace for restaurants & home kitchens and with the other he invented how to properly revenge feminine insults (about which I know nothing but maybe Dr. Martinás knows). The other legacy is that there is not even an approximate balance equation for Heat.

Hence, in another century, it turned out that there is no such quantity as Heat in Thermodynamics. The exact proof involves integrability conditions & such; but for a handwaving argument: Temperature T is Intensive, so Heat Q should be Extensive. But extensives obey Balance Laws (not necessarily Conservation Laws). Now, a cannon being just stationarily bored pours out Heat "from nothing", something definitely not a Balance Law.

Thermodynamics is not yet exhausted. A small but very involved question is treated in [15]; a part of it was separately written and it is [16]. Seemingly there is no problem, we know how to describe the system until no Thermodynamics is meant.

But one problem remained even after such giants as Rumford, Joule, Carathéodory & Ehrenfest-Afanassjeva. OK, there is no Q. Independent extensives are, for example, V, S & N; potential is E. Then the differential First Law is

dE = -pdV + TdS + µdN (1)

On the other hand, there is irreversibility, so

dE = dQ + dW (2)

where dQ is the irreversible part of the change.

And then if there are no Perpetua Mobilia of Second Kind, then you can prove the Integrating Divisor, i.e. that

dQ = TdS (3)

So the Pfaffian's rank is K=2. K=1 would lead to the Mercantilists' Ancient Régime economy having ended in French Revolution, and K=3 would lead to the economic Perpetua Mobilia of Chief Attorney Fouquier-Thinville, asking capital punishment for Lavoisier, of Robespierre, and of others. (I do not prove my last two statements for brevity, but on my homepage you find some related studies). If there are no Perpetua Mobilia, (3) indeed follows.

However, what is exactly dQ? We met this problem after finishing [10]; S is the parameter in foliation, but S is not unique. But the decomposition of dE to dQ and dW is indeed not unique. Really, the dW part is the reversible part. But you can add some parts of dW to dQ. Then the new dW is still reversible and the new dQ is irreversible.

The problem is exposed in [17], but has not yet been explicitly solved. But it will be soon.

8. EPILOGUE

History of Science exposes a nice story about Galileo and the Dominican Lorini. The morale will not be what most readers expect. In October 1612 Lorini delivered a sermon against Ipernicus, whose teaching is not conform with the Bible [5]. Galileo writes a letter; Lorini answers. He indeed believes that this Ipernicus, or what is his name, is not conform with Bible. But he has other things to do, and he preached about Ipernicus only to show that he is there and he too has an opinion.

I also wanted to demonstrate that I am here.

APPENDIX A: BOLYAI, LOBATSCHEWSKY AND PRIORITY

As it is well known (in Hungary), the possibility of non-Euclidean Geometry was invented/proven twice, independently, by Bolyai (the younger) at Marosvásárhely and by Lobatschewsky at Kazan.

Foreigners generally regard Lobatschewsky as sole discoverer, or at least one of tremendous priority. Interested readers can turn to my work on Internet [18]. If one wants only the barest facts, they are told here. If one does not need even the barest fact, skip this Appendix.

Russians generally refer the invention as [19], but they note that there was a previous lecture with the title "Voobrazhaemoi geometrii". Of course nothing is known about this purely oral lecture; and even if it happened, the title means cca. "Imaginary Geometries", so it is hard to imagine anything else than informal discussion of possibilities.

With the written work [19] the problem seems to be that nobody exactly states what is in the article series. It seems that the volume does not exist outside of Russia; Ref. [20] explicitly states that "No copies of Lobachevsky's first publication of his non-Euclidean, hyperbolic, geometry in the Kazan Messenger [Kazanski Vestnik] are known to exist.". While surely some paper did exist, since both Academician Ostrogradskii in 1832, and the angry editorial of journal Syn otechestva (Son of Faterland) in 1834 tell that his work is full with errors [21], both mention a "kniga", so a "book", not a journal. Indeed, Kazanskii Vestnik became the Journal of the University in 1834.

While Kazanskii Vestnik may have existed in 1829/30 as e.g. a periodical of the City of Kazan, surely, we should know what appeared in it in 1829/30. As for the "book", Syn otechestva is not classified to referee Non-Euclidean Geometry. Academician Ostrogradskii from the Gauss-Ostrogradskii theorem may or may not be; he refused the theory, but the problem is, we do not know, exactly what theory he refused.

There is a regular publication from 1840 [22]. However compared to that Bolyai had priority. According to Hungarian History of Sciences, the first ms. was ready in 1826; but that paper has been lost by the military of the Empire, so we cannot know if it was complete. However offprints existed in 1832; Gauss read it just before 6th March 1832. The book of the older Bolyai, to which the theorem was an Appendix, appeared in 2 Volumes, in 1832-33 and generally cited as [23], but the true bibliographic data are [24]. You can compare them. While the two cite the same book, it is rather nontrivial to identify them.

If you can accept a Danish-American physicist impartial, I can cite a sci-fi [25]. Its last-but-one Chapter (in the cited edition the XXXIVth) ends with the sentence: “And this is how it happened that, although Bolyai led our expedition, Lobachevsky published it.” (Of course this is not about the Parallels, but an expedition to Hell, but may be used as analogy.)

APPENDIX B: ON EXTROPY

An idea is to form a quantity roughly in the same relation to entropy as exergy to energy.

I will not go into the details in explaining my present comment; sapienti sat. However one author calls this quantity extropy. Now extropy cannot and do not exist.

This is not a repetition of Ludwig von Bertalanffy's comment that ektropy does not exist. Ludwig von Bertalanffy emphasized that nonexistence because he was not Mazdaist. For all physicists and most scientists it is superfluous to assume two warring tendencies, one for Chaos, another for Order. My statement is grammatical.

"Entropia", coined in late XIXth century, is formally a Classical Greek word, of the same strucure as "energia". Here the root can be seen from the words meaning "work": εργον & εργασια; and εν- is a preposition or such, meaning "in". So, "energia", energy is an "internal capacity for work". If you form something which is the ability "compared to something external", as e.g. in hydroelectric plants, then it is rightly "exergy".

Now let us see entropy. The second half is the root in τροπη, "run", or tropism, "a tendency into some direction". So "en-tropy" is some "Internal Arrow of Time".

Good, now form the term for "the same but external", maybe "for the same in open systems". It cannot be extropy.

Namely, εργα starts with a vowel. True, in Mycenian Greek it started with a "w" [26], but that was then. In Classical Greek the "w" was lost, except in Aeolian and maybe Arcadian. (The letter is the digamma.) However τροπη starts with a consonant. Before a consonant the preposition is not εξ- but εk-; see e.g. εkστασις, so ecstasy, not extasy.

The Greek word εξτροπια is grammatically impossible, so does not mean anything. If it is a brand new English word, then it may mean anything, and it is dangerous to operate with undefined words in Science, while it is quite usual to do it in Art.

REFERENCES

[1] Molnár T. B. & Bittera Dóra: A gróf sparheltja. Magyar Nemzet, 23rd April, 2005. (The title means "The Count's Cooking-range"; a part of a long series about cookery.)

[2] K. Martinás: personal communication, 1988

[3] Adams J. Th.: Amerika eposza. Az Egyesült Államok története. Athenaeum, Butapest, n.d. (This is a Hungarian edition; I am looking for the original.)

[4] Simonyi K.: A fizika kultúrtörtrénete. Gondolat, Budapest, 1978

[5] P. S. Kudryavcev: Istoriya fiziki. Gosuchpezdat, Moscow, 1948

[6] Lengyel B., Proszt J. & Szarvas P.: Általános és szervetlen kémia. Tankönyvkiadó, Budapest, 1959

[7] Marx K.: A tôke. Kossuth, Budapest, 1967. (Of course, this is Hungarian edition. I never saw English. You can look for Ch. Marx: Capital, and that is it. The Rumford Soup is in Chapter 23.)

[8] Horváth F.: A szobafűtés elmélete. Budapest, 1875

[9] J. Robison: Outlines of a Course of Lectures on Mechanical Philosophy. Edinburgh, 1797

[10] B. Lukács & K. Martinás: Callen's Postulates and the Riemannian Space of Thermodynamic States. Phys. Lett. 114A, 306 (1986)

[11] Count of Rumford: The Nature of Heat. Ed. by S. C. Brown, Cambridge Mass. 1968

[12] Szerb A.: A királyné nyaklánca. Bibliotheca, Budapest, 1943

[13] Aristotle of Stageira: The Complete Works, ed. by J. Barnes, Princeton University Press, Princeton, 1995, Bk. N° 198a14-b14

[14] B. Lukács: On the Thermodynamics of Evolution. In: Evolution: from Cosmogenesis to Biogenesis. Proc. 1st Symp. on Matter Evolution, eds. B. Lukács & al., KFKI-1990-50, p. 130; it will be on the Net soon.

[15] Nóra Fáy, B. Lukács, K. Martinás & Sz. Bérczi: On Cosmic Spherule Composition: Gravitation & Thermodynamics. PIECE'99, ed. Miura Y., Yamaguchi, 1999, p. 25.

[16] B. Lukács & K. Martinás: On Thermodynamics of Asteroid Mantles. http://www.rmki.kfki.hu/~lukacs/MARTBAS2.htm

[17] B. Lukács: On Heat Death in Past, Present and Future. Acta Climat. XXIV, 3 (1992). Also on the Net: http://www.rmki.kfki.hu/~lukacs/TIMEARR.htm

[18] B. Lukács: The 200 Years of Bolyai, Construer of Noneuclidean Geometry. http://www.rmki.kfki.hu/~lukacs/BOLYAI.htm

[19] N. I. Lobachevskii: O nachalah geometrii. Kazanskii Vestnik 1829/30, N°'s 25, 27 & 28. (The title means cca. On the Foundation of Geometry.)

[20] ***: The William Marshall Bullitt Collection of Rare Books in Mathematics and Astronomy. http://library.louisville.edu/library/ekstrom/special/bullitt/bullitt.html

[21] ***: Nikolai Ivanovich Lobachevskii (1792-1856). http://translit-www.klax.tula.ru/~volodin/lobachevsky.html

[22] N. I. Lobatschewsky: Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin, 1840

[23] F. Bolyai: Tentamen ..., Maros Vásárhely, 1832-33. (The respective Appendix is: J. Bolyai: Appendix Scientiam Spatii ...)

[24] Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Cum Appeddice triplici. Josephus et Simeon Kali de felsô Vist. Auctore Professore Matheseos et Physices Chemiaeque Publ. Ordinario. Maros Vásárhelyini. 1832, 1833. I will not give here a word for word translation (look, this is not Magyar, but international Latin), but explain the sentences. Sentence 1 is the Title. Sentence 2 states that 3 Appendices are added. Sentence 3 gives the publishers. Sentence 4 circumscribes the author (Math, Phys. & Chem. Prof; for any reason F. Bolyai did not want his name on the book!). Then comes the place of the publication: the Latin locative of the Magyar name Maros Vásárhely. If one finds this book after the citation [23]... Then one of the 3 Appendices is the work of J. Bolyai. Its title is: Scientiam spatii absolute veram exhibiens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica. I must admit that the Bolyais did not help the popularity of this Theorem, and in contemporary West only Latin linguists & Catholic priests could have been interested. However: this was both the official and working language of the Hungarian Parliament(s) & county assemblies.

[25] P. Anderson: Operation Chaos. Orb, New York, 1999. Originally the story was told in 4 short stories: this is in Operation Changeling.

[26] J. Chadwick: The Decipherment of Linear B. Cambridge University Press, Cambridge, 1958

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