Elemér Kiss: Mathematical gems from the Bolyai chests

(extended 2nd edition)



“Whatever I could say about this book can already be found in the book itself.”

-Albert Einstein replied to a reporter of the New York Times who asked him about his book written with Infeld. This answer is not only witty but especially true to every book which reveals revolutionary ideas in a given field. The reason for it being that real novelty cannot be compared with or described by simple references to already known concepts.

The task Elemér Kiss took on, deciphering the contents of the “Bolyai-chests” (thousands of handwritten letters and manuscripts) led to extraordinary results.The expression’deciphering’ shows the tedious act of many decades by which it has been possible to reconstruct the contents of these materials. The contents, the grammar, the mathematical symbols of which differed significantly from that of present times’ and were often unreadable. Today we know that the results of this hard work left us with a brand new picture of Janos Bolyai.

The book “Bolyai-chests” (published in 1999) is the systematic but not comprehensive elaboration of Bolyai’s heritage by Elemér Kiss which was published in English and Hungarian by Typotex and Akademia Publishers. In the six years since then, all the copies of the 1st edition have been sold which made this 2nd edition with extended contents necessary.


The 2nd extended edition has a similar structure to the first one. In addition, each chapter has been enriched by corrections, new connections and critical reviews of the literature that has been published since 1999.

The 1st chapter ‘The life of Janos Bolyai and the science of space’ gives a brief account of the journey the scientist took to creating a new geometry. In addition there is a real novelty in Chapter 1.6 considering the creation of non-Euclidean geometry based on facts from Bolyai’s correspondence. The author comes up with a convincing reasoning for the priority of Bolyai in the Bolyai-Gauss-Lobacsevszkij relation.

In Chapter 2. we can read a systematic and comprehensive description of the “Bolyai-chests”. Parts of this chapter explain the language and symbols used by Bolyai, which is the result of meticulous research, as some of the original texts resemble complicated riddles.

Chapter 4. that offered numerous exciting experiments even in the 1st edition has now been extended by subchapters 4.9 “Music and mathematics” and 4.10 “J. Bolyai and the Diophantical equations”.

These thoughts were recorded by Bolyai in the 1840s when he was also working on his study called ‘Music Theory’.

Chapter 4.6 represents special value where Bolyai’s theorem on Fermat numbers gets introduced. According to this “all Fermat numbers take the form of 6k-1”. The international significance of this theorem and the weight of Elemér Kiss’s research alike have been proven by the publication of ’17 Lectures on Fermat Numbers’ by Krizek-Luca-Somer (Springer Publishers) in 2004 where this theorem was called ‘Bolyai Theorem’. The study had originally been published in 1999 in Historia Mathematica.

This is the first, highly-reputable source where Janos Bolyai’s name is mentioned in the field of number-theory as opposed to geometry, which is a real milestone on the way to revealing the real face of Janos Bolyai.

Chapter 5, the “Theory of Primes” talks about a few results of Bolyai’s work that has never been seen before: about Complex Integers, into the research of which Bolyai invested huge amounts of energy. These studies that Bolyai called the ‘theory of primes’ dealt with the arithmetics of complex integers.

Chapter 6, “The Theory of Algebraic Equations” reveals Bolyai’s struggles in connection with the solvability of algebraic equations of fifth and higher order. It has been summed up by Elemér Kiss at the end of this chapter: “Janos Bolyai thought long about this important problem without knowing that it had been resolved before. On the other hand, the world didn’t know about this 19th century Hungarian scientist who –perhaps late and only for its own sake- had put an end to a centuries long debate.

I want to emphasize that the importance of this quotation is in the wording: “thinking long about this important problem” rather than “algebraic problem”. This suggests the isolation Bolyai worked in all his life, and the enormous creative power through which he was able to “make up a new, different world out of nothing”, not excusively in the field of geometry.

A brand new 7th Chapter has been added to the original edition, with the title “J. Bolyai’s research in the area of analysis”. What makes it really interesting is what we learn from a letter written by Farkas Bolyai (Janos’s father) to Gauss in 1816. “My son … likes differential and integral calculus and counts with them with ease and pleasure.”

In the 8th Chapter we find Bolyai’s letters which is invaluable, considering that they have been ‘translated’ into a language easily digestable to the modern reader.

To complete the understanding of the mathematician’s work, in the 9th Chapter the “Terminology and Symbols used by Bolyai” can be found.

At the end of the book there is an extended Bibliography containing 166 items.


It is hard to overestimate the value of Elemér Kiss’s work in the process of revealing Janos Bolyai’s real face after 150 years of silence. What makes it even more meaningful is the fact that Janos Bolyai’s face has been unknown so far, as the famous image that has been circulated around the world is certainly not his. To put an end to this misconception and also to publicize essays and research in related areas “The Real Face of Janos Bolyai” has been created under the address www.titoktan.hu



Budapest, February 2006.                                                                                                                                                                                       T. Dénes T.