*I cannot say more, only that from nothing I have created a new different world*

János Bolyai, letter to Farkas Bolyai describing his discovery of non-Euclidian geometry3 November 1823__[1]__

Of all the enduring mysteries of life, the nature of space stands pre-eminent. To know whether our cosmos extends infinitely devoid of curvature, or whether it unfurls to the gentle arcature of a saddle might appear an esoteric indulgence, but humanity has long demonstrated a profound thirst for resolving this mystery.

Sadly, our pursuit of knowledge has often been hindered by failures of communication among scientists.

This essay examines a particularly lamentable example of such a failure. Exploring the limited interactions between János Bolyai, a brilliant 19th-century Hungarian mathematician, and the renowned Carl Friedrich Gauss, this essay will demonstrate how a matter as simple as a letter that the recipient found offensive was enough to rob the world of thousands of pages of illustrious mathematical thinking, stalling progress in the fields of geometry, mathematics and number theory for decades if not longer.

This essay will also address the historical context of their interactions. It will also address the importance of clear communication in the sciences, as well as the broader consequences of their failed epistolary exchange. We will also draw parallels between the paradigm shifting communications between Werner Heisenberg, Wolfgang Pauli, and Niels Bohr only a century later. In doing so, the essay will highlight how essential the sharing of ideas and findings is to the collective growth of scientific understanding.

**An Epistolary Failure of Great Consequence**

From Sumerians with their primitive astrolabes__[2]__ to the sophisticated European Euclid Space Mission of 2023, humanity has spared little effort in its endeavor to understand the nature of space.

The discovery of *non-Euclidian geometry*, the study of curved surfaces, is widely regarded as one of humankind’s greatest mathematical feats,__[3]__ not least because of the immense implications it has had for our understanding of the space. During the first half of the 19th century, three mathematicians, Nikolai Ivanovich Lobachevsky, Carl Friedrich Gauss and János Bolyai, would independently develop mathematical frameworks for non-Euclidian geometry.

While all three would be connected through mutual friendships and acquaintances,__[4]__ fate conspired to prevent any direct communication about their findings. To make matters worse, the communications that did occur between them would serve to hinder the progress of science.

János Bolyai would be the first to write down his framework in 1823.__[5]__ He would publish it, with the help of his father, a decade later. While Lobachevsky would be connected through mutual acquaintances, he would never learn of Bolyai’s work before publishing his own framework in several years after Bolyai.__[6]__ Gauss, the most eminent figure of the three, __[7]__ would receive a copy of Bolyai’s findings, but he himself never published on the topic.

Through his response to the copy he received, Gauss would, however, offend Bolyai__[8]__ to such a degree that he would never publish a single page again.

As a result, the communication failures of Bolyai, Gauss and Lobachevsky would not only stall progress in non-Euclidian geometry but also deprive the world of an entire corpus of deeply inspired mathematical writing which Bolyai would decide to hide from the world.

**János Bolyai: A Demon Sprung Upon the Field of Mathematics and Geometry**

János Bolyai was born on 15th December 1802 in Hungary to mathematician and professor of philosophy at the College of Kolozsvár Farkas Bolyai and his wife Zsuzsanna Benkö. The younger Bolyai would prove to be a mathematical genius par excellence. In addition, he demonstrated remarkable talent in a wide range of pursuits from playing the violin to the martial art of dueling which he perfected during his lengthy military career.__[9]__

His most enduring legacy was to be in mathematics, a discipline to which he ‘*sprang like a demon*’,__[10]__ as his father attested. Bolyai’s independent discovery of non-Euclidian geometry at the tender age of 21__[11]__ was a feat of extraordinary magnitude, not least because it resolved one of the longest standing mathematical problems concerning Euclid’s axioms which we will discuss below.

Bolyai would present his findings in a 26-page appendix to his father’s mathematics textbook, *Tentamen*,__[12]__ in 1832. From there, he would continue to produce more than 10,000 pages of mathematical manuscripts that spanned algebra, number theory, __[13]__ and the elucidation of Fermat's theorem on primes__[14]__ to name a few.

None of these following pages would be published.__[15]__ In fact, little would be known of how far ahead the rest of the scientific community he was without the work of Paul Stäckel at the turn of the 20th century__[16]__ and the effort of modern mathematical historians who have explored Bolyai’s unpublished corpus over the past decades.__[17]__

To give a sense of the magnitude of the loss Bolyai’s withdrawal caused, consider the fact that his appendix to *Tentamen*would be cited as the “*most glorious 26 pages of Hungarian science*.”__[18]__

What drove Bolyai to reject the scientific community is all contained in a solitary letter to Bolyai senior from Gauss. Before we explore the exchange in more detail, it behooves us to examine the scientific milieu in which Gauss and Bolyai toiled.

**A Flat World Rendered Asunder**

From 300BC to the Enlightenment, Euclid’s *Elements*__[19]__ and the 13 books it comprises were the defining treatise of mathematics and plane geometry. In Elements, Euclid lays the foundations for the entire field of geometry through five axioms__[20]__ from which every other feature, notion and geometrical outcome would be derived.

Of these axioms, it is the fifth and final one, *the parallel postulate*,__[21]__ that Bolyai would ultimately dismantle in a revolutionary stroke of genius.__[22]__

Generations of mathematicians before Bolyai, such as Girolamo Saccheri and Johann Heinrich Lambert, had attempted to formally prove Euclid’s fifth axiom without success.__[23]__ What Bolyai demonstrated was that the parallel postulate was not a strict necessity for creating a consistent mathematical framework of geometry and space.__[24]__ Instead, a geometry where parallel lines intersect is possible, as long as we boldly dismiss Euclid’s fifth axiom entirely.

In 1823, Bolyai stood out as the solitary mathematician who had recorded his attempts at disproving the parallel postulate. In doing so he created a mathematical framework for future generations of geometers, physicists and mathematicians to explore the very the nature of space, gravity and much more.

Bolyai himself seemed aware of the significance of his findings when in one autumn letter he wrote to his Father that “*I cannot say more, only that from nothing I have created a new different world*”.__[25]__

**The Words That Would Bend Space**

Bolyai senior was a close acquaintance of Gauss.__[26]__ Upon the publication of the 1832 version of *Tentamen* that included his son’s appendix on non-Euclidian geometry, Bolyai senior dispatched a copy to Gauss in a bid for his reaction.

Unbeknownst to the Bolyais, Gauss had also begun to dream of a new and different world where parallel lines could intersect. On 6 March 1832, Gauss responded to Bolyai senior with a letter that discussed János Bolyai work as follows:

“*If I start by saying “I cannot praise it” then you will most likely be taken aback; but I cannot do otherwise; to praise it would be to praise myself; the entire contents of the work, the path that your son has taken and the results to which it leads, are almost perfectly in agreement with my own meditations, some going back 30 – 35 years. In truth I am astonished*.”__[27]__

The response thoroughly dejected János Bolyai,__[28]__ who would never publish again.

Communications in science presents itself with two faces: one oriented towards the external world and its laymen, the other staring deep into the eyes of contemporary, past, and future peers. That scientists succeed in the latter is an existential concern to all of us. In fact, communication between scientists is *in itself progress in science*. The ways in which the acts of communication can happen are myriad, ranging from letters to colloquia, conferences and in-person cooperation. What matters most is that the communication happens. Ideally, it would happen in a timely, open and respectful manner so that scientists can effectively build a shared knowledge base to leverage in their work.__[29]__

Consider for a moment the alternative; a world where scientists toil away on their lonesome, learning of each other’s work and discoveries only by accident if at all.

Such would be a world of persistent stagnation punctured only by sudden saltations of scientific development whenever a once-in-a-generation genius was lucky enough to push our epistemological boundaries; a world, which János Bolyai, Lobachevsky and Gauss inhabited when working on their frameworks for non-Euclidian geometry.

An Example of Communications and Collaboration as a Driver of Scientific Progress

A bevy of prior literature has examined the intersection of communications, science and history. From examining the birth of the scientific article in the 17th century__[30]__ to exploring Newton’s exchange of letters with his peers__[31]__, the historical discourse shows that communications between scientists has often been a critical driving force of scientific progress.

One of the greatest examples of the transformative power of open communications and collaboration between scientists is the exchanges between Werner Heisenberg, Wolfgang Pauli, and Niels Bohr__[32]__ whom would come to build the entire field of modern quantum mechanics.

Heisenberg, born in 1901, would mirror Bolyai’s precocious productiveness by independently creating the foundations for modern quantum mechanics by the age of 26.__[33]__ Heisenberg's renowned *uncertainty principle*__[34]__ emerged from his work to formulate a precise mathematical framework way to represent the quantum states and energies of electrons. However, it would be misguided to represent the accomplishment as being Heisenberg’s alone.

Shortly after having made his initial discovery in February 1927, Heisenberg wrote a letter to Wolfgang Pauli__[35]__ who was a distinguished professor of physics at the University of Hamburg at the time. In stark contrast to what occurred between Gauss and Bolyai, Pauli not only welcomed Heisenberg’s results but also worked closely with him__[36]__to present the world with Heisenberg’s work through an article that was published in March 1927.__[37]__

Heisenberg also collaborated closely with Bohr, who was decades his senior and widely considered a leading figure in their field.__[38]__ Where the letters between Heisenberg and Pauli were mostly technical in their orientation and at times brutal in its criticism,__[39]__ the exchanges between Heisenberg and Bohr were markedly more philosophical and more conceptual in nature.__[40]__ The trio’s exchange of letters and ideas between the three continued for decades, resulting in more than 800 pages of writing__[41]__ between Pauli and Heisenberg alone.

The collaboration between these three scientists would prove instrumental. Without the mentorship, guidance and constructive critique of his elder peers, it seems unlikely that Heisenberg’s discovery would have blossomed into the foundations of what is now known as the Copenhagen interpretation of quantum mechanics.__[42]__

Truly, successful communication between scientists can change both the world and our understanding of it.

**The World That Could Have Been**

Let us now turn our attention back to Bolyai whom we left despondent in 1832 after Gauss’s seeming dismissal of his greatest discovery.

Where Heisenberg conversed with Pauli and Bohr directly, Bolyai never interacted with Gauss or Lobachevsky himself. Gauss would come to know of János Bolyai’s work only because of Bolyai senior’s persistence in sending copies of *Tentamen* and its appendix over several years.__[43]__ Bolyai senior is also to thank for relaying Gauss’ response to János.

In assessing the failures of this particular epistolary exchange, it is critical to note that while János Bolyai most certainly read Gauss’ response, he was not the intended recipient. Neither was the letter written in his native language of Hungarian. Knowing what we know of young and prideful egos, and acknowledging the subtleties of translating Gauss's letter from German to Hungarian, it becomes impossible to dismiss the notion that Bolyai may have simply misinterpreted Gauss’ response.

True, Gauss stated that he could not praise Bolyai’s work. However, Gauss also exclaimed that “*In truth I am astonished.*” Gauss also drew direct parallels between the work of himself and the younger Bolyai – an act that could just as well be flattery as well as dismissal.

A letter Gauss wrote to his friend Christian Ludwig Gerling on February 14, 1832 gives further credence to this alternative interpretation of Gauss’ initial response. In this letter, Gauss explains that “*I regard this young geometer Bolyai as a genius of the first order*”.__[44]__

One can only imagine the scientific progress that could have taken place had János Bolyai been privy to the contents of Gauss’ letter to Gerling. Perhaps the world would still be reaping the fruits of an intellectual partnership between Gauss and Bolyai that would have transformed an unpublished corpus into staples of science. Perhaps we would be recounting the collaboration between Gauss, Bolyai and Lobachevsky instead of Heisenberg, Bohr and Pauli as the prime example of success in communications between scientists.

But alas, that world would never come to pass, and the world that could have been will forever remain a letter’s width away from our reach.

T. Alexander Puutio is currently undertaking an MSt in History at the University of Cambridge (Wolfson College).

**Notes:**

__[1]__ János Bolyai, ‘Temesvár Letter from János to Farkas Bolyai’, Translated by Péter Körtesi, November 3, 1823. School of Mathematics and Statistics, University of St. Andrews, Scotland, MacTutor. https://mathshistory.st-andrews.ac.uk/Extras/Bolyai_letter/.

__[2]__ G. Çağırgan, ‘Three More Duplicates to Astrolabe B’, *Belleten*, Vol. 48, No. 191-192 (1984), pp. 399-416.

__[3]__ George Bruce Halsted, ‘Gauss and the Non-Euclidean Geometry’, *The American Mathematical Monthly*, Vol. 7, No. 11 (1900), p. 247.

__[4]__ J.J. O’Connor and E.F. Robertson, ‘Nikolai Ivanovich Lobachevsky’, *MacTutor History of Mathematics Archive*, __https://mathshistory.st-andrews.ac.uk/Biographies/Lobachevsky/__.

__[5]__ George Bruce Halsted, ‘Biography: John Bolyai’, *The American Mathematical Monthly*, Vol. 5, No. 2, 1898, pp. 35–38.

__[6]__ Valentin A. Bazhanov, ‘Nikolay Ivanovich Lobachevsky’, *Encyclopedia Britannica* (2023), https://www.britannica.com/biography/Nikolay-Ivanovich-Lobachevsky.

__[7]__ W.K. Bühler, ‘Gauss: A Biographical Study’, *Springer-Verlag* (1981).

__[8]__ ‘Bolyai János’, *Slovak University of Technology in Bratislava* (n.d.).

__[9]__ Halsted, ‘Biography: John Bolyai’.

__[10]__ Ibid.

__[11]__ Ibid.

__[12]__ János Bolyai, ‘Appendix Explaining the Absolutely True Science of Space’, Farkas Bolyai (ed.), *Tentamen* (Transylvania, 1832),

__[13]__ See e.g. Róbert Oláh-Gál and Alexandru Horvath, ‘Deep Geometrical Thoughts from Some – Until Now Not Published – Manuscripts of János Bolyai’, *Proceedings of the 3rd Conference on the History of Mathemetics and Teaching of Mathematics, University of Miskolc* (2004), pp. 65-75.; and Elemér Kiss, *Mathematical Gems from the Bolyai Chests: János Bolyai’s Discoveries in Number Theory and Algebra as Recently Deciphered from His Manuscripts* (Budapest: Akadémiai Kiadó, 1999)

__[14]__ Elemér Kiss, ‘Fermat's Theorem in János Bolyai's Manuscripts’, *Mathematica Pannonica*, Vol. 6, No. 2 (1995), pp. 237-242.

__[15]__ Morris Kline, *Mathematical Thought from Ancient to Modern Times* (Oxford: Oxford University Press, 1972)

__[16]__ Paul Stäckel and Friedrich Engel, *Die Theorie der Parallellinien von Euklid bis auf Gauss; eine Urkundensammlung zur Vorgeschichte der nichteuklidischen Geometrie* (Leipzig: B.G. Teubner, 1895)

__[17]__ ‘Bolyai János’, *Slovak University of Technology in Bratislava.*

__[18]__ Elemér, *Mathematical Gems*.

__[19]__ Euclid, ‘Elements’, Translated by Hypsicles of Alexandria (Venice: Erhard Ratdolt, 1482)

__[20]__ D.M.Y. Sommerville, *The Elements of Non-Euclidean Geometry* (2005)

__[21]__ R. Ravindran, ‘Euclid’s Fifth Postulate’, *Resonance*, Vol. 12 (2007), pp. 26-36.

__[22]__ A. Prékopa, ‘The Revolution of János Bolyai’, in A. Prékopa and E. Molnár (eds.), *Non-Euclidean Geometries: János Bolyai Memorial Volume: *581 (New York: Springer, 2006)

__[23]__ Jeremy Gray, *In: Worlds Out of Nothing*: *A Course in the History of Geometry in the 19th Century* (New York Springer, 2010)

__[24]__ János Bolyai, ‘Appendix Explaining the Absolutely True Science of Space’, in Bolyai, *Tentamen*.

__[25]__ Bolyai, ‘Temesvár Letter’.

__[26]__ J.J. O’Connor and E.F. Robertson, ‘Farkas Bolyai’, *MacTutor History of Mathematics Archive *__https://mathshistory.st-andrews.ac.uk/Biographies/Bolyai_Farkas/__.

__[27]__ Halsted, ‘Gauss and the Non-Euclidean Geometry’, p. 247.

__[28]__ Oláh-Gál and Horvath, ‘Deep Geometrical Thoughts’.

__[29]__ National Academies of Sciences, Engineering, and Medicine, Communicating Science Effectively: A Research Agenda, Division of Behavioral and Social Sciences and Education, Committee on the Science of Science Communication, ‘Building the Knowledge Base for Effective Science Communication’, *National Academies Press* (2017).

__[30]__ Gross, Alan G., Joseph E. Harmon, and Michael Reidy, *Communicating Science: The Scientific Article from the 17th Century to the Present* (Oxford: Oxford University Press, 2002)

__[31]__ H.W. Turnbull (ed.), *The Correspondence of Isaac Newton: 1661-1675 *(Cambridge: Cambridge University Press, 1959)

__[32]__ See: Wolfgang Pauli, *Wissenschaftlicher Briefwechsel Mit Bohr, Einstein, Heisenberg U.A. Bands I-IV 1919-1954 *(Berlin: Springer Berlin, Heidelberg 1979-1999)

__[33]__ David Cassidy, *Uncertainty: the Life and Science of Werner Heisenberg* (New York: Palgrave, 1992)

__[34]__ Jan Hilgevoord and Jos Uffink, ‘The Uncertainty Principle’, *The Stanford Encyclopedia of Philosophy* (2023) __https://plato.stanford.edu/entries/qt-uncertainty/__.

__[35]__ ‘February 1927: Heisenberg's Uncertainty Principle’, *APS News*, Vol. 17, No. 2 (2008) __https://www.aps.org/publications/apsnews/200802/physicshistory.cfm__.

__[36]__ Thayer Watkins, ‘The Drama in the Development of Quantum Mechanics in 1926-27’, *San José State University*__https://____www.sjsu.edu/faculty/watkins/quantumdrama.htm__.

__[37]__ W. Heisenberg, ‘Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik’, *Zeitschrift für Physik*, Vol. 43 (1927), pp. 172-198.

__[38]__ Arne Schirrmacher, ‘Bohr's Atomic Model’, in Daniel Greenberger, Klaus Hentschel, and Friedel Weinert (eds.), *Compendium of Quantum Physics* (New York: Springer, 2009)

__[39]__ Cecilia Jarlskog, ‘Correspondence With Heisenberg Through Pauli’ in Cecilia Jarlskog (ed.), *Portrait of Gunnar Källén: A Physics Shotting Star and Poet of Early Quantum Field Theory* (New York: Springer 2014).

__[40]__ Jagdish Mehra, ‘Niels Bohr's Discussions with Albert Einstein, Werner Heisenberg, and Erwin Schrödinger: The Origins of the Principles of Uncertainty and Complementarity’, *Found Phys*, Vol. 17 (1987), pp. 461-506.

__[41]__ Cassidy, *Uncertainty.*

__[42]__ Carlo Rovelli, *Helgoland: Making Sense of the Quantum Revolution* (London: Penguin, 2021)

__[43]__ ‘Life of János Bolyai’, *Hungarian Academy of Sciences* (n.d.)

__[44]__ J.J. O’Connor and E.F. Robertson, ‘János Bolyai’, *MacTutor History of Mathematics Archive *__https://mathshistory.st-andrews.ac.uk/Biographies/Bolyai/__.

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