Bolyai's Appendix(this is an ongoing chapter)
Bolyai Janos(1802-1860) is one of three founders
of non euclidian geometry．
- Born in Klausenburg in Transylvania（currently
in Rumania，historically in Hungary before
occupied by Austria）．Spent young days in
Maros-Vaserhely in the same region.
- Father, Bolyai Farkas,studied mathematics
in Gettingen when Gauss was also there．They
often talked about math especially Euclid's
- Discovered non Euclidian Geometry around
23 years old. Have it published in 1831 as
an Appendix of his father's math book ,Tentamen．Sent
it to Gauss.
- Gauss appreciated the paper very much within
his circle but did not say anything about
the paper in public.
- Without any recognition from the public,
he became ill. Retired from the Austrian
army and started living on pension in 1933.
- He had three children with a common-law wife
Rosalie. Their status of not formally getting
married and the different idea about math
lead Bolyai away from his father. His father
died in 1856. Around the same time, he got
also apart from Tosalie.
- Died at 58 years ols in 1860. The body was
buried in a public cemetery near a church.
A record of the church said "he died
Almost all info above , I got from 'Janos
Bolyai' by Jeremy J. GRAY(2004) except the
last quote which I took from a book "isetu
sugakusha retuden" by mori tuyoshi(japanese
mathematician). Galoia is very famous
for his tragic death(life), but Bolyai can
almost match him. （He should have lost this
His first book is 'Appendix' in 1831 in Latin,
later he translate it into German. That book
is RaumLehre, which is almost same as Appendix
until section 31, but cut short from chapter
32. I read RaumLehre in Japanese translation
by prof. Terasaka. He translate RaumLehre
until section 31 and translate Appendix from
When I was facinated by non euclidian geometry
a few years ago, I started from models(upper
sphere model & H+ model) . It was fun
because I could enjoy strange geometry and
programing also. But after reading Bolyai,
I was really impressed by his strong ideas
and geographical images. He is like
a wizard. He draws unbeliebale lines or even
surfaces in the middle of nowhere to me.
To probe a theory in a plane(2D) , he
sometimes draw a surface and do it in 3D.
Before reading his, I had read Lobachevsky
and he also use 3D for 2D proof, but his
style is too elegant to understand for me.
While Bolyai's style seemingly is more of
an amature and might lack solid proof sometime,
but anyway I love his intuitive way of proof.
In addition to that, I am interested in his
life too. He was a very passionate mathmatician
not only in math but also violin and fencing.
One day he is said to win a dozen of opponents
drinking wodka! Some day I would like to
visit Transylvania. Anyway these are the
reasons why I started this page.
Please refer to 'Appendix' to read the
proves. There is an English translation.
You can find it in such as
'Janos Bolyai' by Jeremy J. GRAY(2004) ,
'Non Euclidian Geometry' by BONOLA
But not all versions of BONOLA have the translation.
Please make sure you buy the right version.
In this chapter, "AB|||CD" means
AB parallel to CD (they have one infinite
point. And AB is not necessarily a segment/line.
It can be equi distant line or horicircle).
"AB" means circumferance of a circle
centered at A and passes B, while AB means inside of or square AB．L means horicircle, F means hori sphere. These names come from
Linie（line)& Flache(surface) in German.
They are adopted by Bolyai.
Almost all files are written by Geogegra
3.2 & Cabri3D. To manupulate Geogebra
files embedded in htmls , you need Java environment
(JDK/JRE). You can download JRE from here. To manupulate Cabri3D files, you need plugins
for Cabri3D, even if you have Cabri. To download
the plugin, check these: cabri2 , cabri3D ． To change the viewpoint in Cabri3D files,
drag a mouse with the right buttom down.
At the bottom, there are also original
Geogebra&Cabri3D files which you can
directly manipulate with Geogebra/Cabri3D
SOFTWARE (not plugins)．
If you can read Japanese , you can read detailed explanations of these
files and mathematical proves from here. They are PDF files with one to one links
to Cabri/Geogebra files.
When you read PDF files and want to return
from a link, please use the return key of
the browser!! If you use the close button,
browser will close.
ogose shigeki(), all rights reserved