Elemér Kiss: Mathematical Gems from the Bolyai Chests


JÁNOS BOLYAI is one of the greatest characters of both Hungarian and universal science. His life and works have been discussed by numerous authors; fortunately, they wrote much about him. The literature on him, thanks to the diligence of our precursors, could now fill a whole library. Noteworthy monographs, a number of scientific papers and popular articles trace his course of life, his landmark discovery in geometry and his role in shaping modern mathematics.

JÁNOS BOLYAI the mathematician has been granted a place in science worthy of him a long time ago. Not only Hungarians, but in fact the whole world knows and esteems the name BOLYAI. Astronomers named a crater on the Moon after him, as well as after other Hungarians: MIKSA HELL, LORÁND EÖTVÖS and JÁNOS NEUMANN. One of the asteroids orbiting between Mars and Jupiter also bears his name. When the first Hungarian astronaut entered outer space, he carried with him a copy of the original edition of the main work of JÁNOS BOLYAI. He thus indicated that Appendix constituted the most glorious 26 pages of Hungarian science.

Whenever his name is mentioned, it is his results in geometry that are emphasized. This is certainly justified, since his creation of absolute geometry is truly an unequalled achievement. Recently, more and more attention has been paid to his social and philosophical views, and perhaps his Responsio is also becoming known in the history of mathematics. There exists, however, one lesser-known, nonetheless important work of JÁNOS BOLYAI: his manuscripts left to us after his death. The thousands of pages of the Bolyai relic contain those ideas that their writer recorded in the second half of his life, living in lonesome and dismal surroundings.

The processing of BOLYAI's heritage started about 100 years ago. The scientific world got acquainted with several manuscripts of JÁNOS BOLYAI, thanks to the pioneering work of PAUL STÄCKEL published around the turn of the century. Ensuing studies of BOLYAI were mainly based on the observations and explanations of STÄCKEL. His study is indeed significant, but he could neither take into account nor properly evaluate all aspects of BOLYAI's work appropriately. His book, in spite of occasional inaccuracies and errors, is the most detailed and thorough monograph on BOLYAI to date.

Can anything new be said about the author of Appendix? In 1997, is now, our knowledge about BOLYAI deficient in any way? These are some of the questions that come straight to mind.

Is it true that JÁNOS BOLYAI was a hundred percent geometrician and that all his original ideas were of a geometrical nature? The authors of the monographs published so far have mentioned that he "aimed to" or "expected to" come up with an answer to certain non-geometrical problems. They have not, however, answered the question whether BOLYAI searched for and found answers to the problems set by himself.

I was urged by the feeling that it was necessary to explore our cultural and scientific heritage, and, at the same time, to make the portrait of BOLYAI more complete and truthful. This urge led me to read through and decipher the thousands of pages of the Bolyai manuscripts at the Teleki-Bolyai Library of Marosvásárhely, and to investigate the results of the creator of absolute geometry in other fields. I could not, for the time being, venture to present all the problems examined by BOLYAI. So far I have been focusing only on those parts of his research that were directed towards certain problems of algebra and number theory. These writings are truly fascinating, and I am confident that, as a reward of my steadfast and meticulous investigation, I have succeeded in disclosing some new and hitherto unknown aspects of the "Bolyai chests". These writings have revealed a picture of BOLYAI more detailed, complete and colourful than the one we used to have.

In this book, supplementing my previously published studies, I would like to report on those discoveries of JÁNOS BOLYAI which lay hidden in his collection of manuscripts and which the rather vast literature on him is entirely unaware of. My objective is to fill some of the gaps still existent in our knowledge about BOLYAI, and to encourage others to do similar research. I have tried to put right some of the errors of earlier studies, and to support all my arguments with the words of JÁNOS BOLYAI, taken from still unpublished documents.
In the course of my work, whenever I could, I refrained from using secondary and tertiary sources, and acquired information only from primary ones, namely, the manuscripts. In many cases it has proven to be useful to contrast other authors' assertions with the original writings.

It is true that no work comparable to the Science of Space can be found in the heritage, but if one has the patience to dig deep enough, one can unearth many surprises that the Bolyai manuscripts still have in store. The efforts are richly rewarded by those treasures and pearls of thought that can be brought to the surface. I sincerely hope that the reader will consider as such JÁNOS BOLYAI's observations stored in Teleki Library. These ideas had remained hidden from the mathematical world, until they were discovered again and published in different journals by others, decades after the death of this brilliant mathematician.

On the pages of his manuscripts, JÁNOS BOLYAI quotes his father's opinion about writing as thus: "... As my father has it, there may be nowadays more writers than readers, and perhaps those deserve mentioning who, being able to read and write, do not in want of a considerable and weighty reason become a writer ... " Although I fully feel the significance of FARKAS BOLYAI's wise words and have considered them at length, I decided, after much pondering and hesitation to try to become a "writer". I believe that I do have a "considerable and weighty reason" to do so.

In recent years I have been scraping together JÁNOS BOLYAI's scattered notes and slips of paper. Thus I have managed to bring to light many of his thoughts which deserve to be developed into full-fledged studies, even if belatedly. BOLYAI, having had a hard fate, could only see 26 pages of his own work in print. Therefore, in what follows, I will often let his own words speak for themselves. I hope that this way we can jointly deliver to the reader at least a portion of what he once wrote down, but could never publish.

I wish to thank all those who encouraged my research and stood by my side with sympathy and readiness to help.

I had the opportunity to examine the autographic Bolyai papers in the Teleki Library. I am grateful to all the librarians for their constant help. Throughout my research, I felt that they were following my work with true sympathy, and that they were delighted at the results achieved.

I wish to acknowledge the encouragement of those editors who first accepted my manuscripts for publication. It was GYULA STAAR, the editor-in-chief of Természet Világa [ World of Nature] and GYULA I. MAURER, editor-in-chief of Mathematica Pannonica who provided support right at the beginning. The grant of Domus Hungarica Scientiarum et Artium, offered as part of the academic programme Hungarian Scholarship Abroad, also gave a decisive push to my work. It made it possible for me to spend some time in Budapest and continue my research there. I owe special thanks to DR. JÁNOS SZENDREI and DR. LÁSZLÓ VEKERDI, the reviewers of the manuscript. Their remarks have greatly helped in making this book more valuable.

Marosvásárhely, 15th December 1997,
on the 195th anniversary of the birth of JÁNOS BOLYAI.