**Elemér Kiss:** **Mathematical gems from the Bolyai chests **

(extended** **2^{nd} 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 1^{st}
edition have been sold which made this 2^{nd} edition with extended
contents necessary.

The 2^{nd} 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 1^{st} 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 1^{st} 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 19^{th} 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 7^{th} 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 8^{th} 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 9^{th} 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