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János Bolyai: Appendix,
Scientiam Spatii
absolute veram exhibens
Maros-Vásárhelyini, 1832
Inscribed 2009
What is it
The Appendix contains the description of János Bolyai’s
revolutionary discovery of non-Euclidean and the more
general absolute geometry, derived from the solution to
an ancient Greek mathematical problem in connection
with parallel lines.
Why was it inscribed
Bolyai’s non-Euclidean geometry changed thinking
about geometry, led to the development of modern
mathematics and paved the way for modern physical
theories of the 20th century. The Appendix records one
of the most significant mathematical discoveries of
the 19th century.
Where is it
Library of the Hungarian Academy of Sciences,
Budapest, Hungary
János Bolyai (1802–60) received his mathematical
education from his father, Farkas Bolyai, also a worldfamous
mathematician, professor at the Reformed
(Church) College of Marosvásárhely, Transylvania. János
first reported his discovery in a letter written to his father
on 3 November 1823, when, as an army engineer, he was
assigned to the Directorate of Fortification of Temesvár,
Transylvania. At about the same time as Bolyai published
his results, the Russian mathematician N.I. Lobachevskii
also published similar results, so the discovery of non-
Euclidean geometry is attributed to both of them.
However, absolute geometry is Bolyai’s sole discovery.
The Appendix, Scientiam Spatii absolute veram exhibens
(‘Appendix, explaining the absolutely true science of space’)
is the primary document of the epoch-making discovery
of János Bolyai. For over 2000 years many of the best
mathematicians tried to prove Euclid’s parallel postulate
(or axiom). János Bolyai created entirely new settings for
the problem by inventing absolute (or neutral) geometry
that is independent of parallelism. He also showed that
348 János Bolyai: Appendix, Scientiam Spatii absolute veram exhibens
no proof was possible and, by assuming all the axioms for
absolute geometry and replacing the axiom of parallelism
by its negation, obtained another geometry of equal
standing to Euclidean geometry. This discovery stimulated
not only the creation of new space concepts vital for
modern physics but, by the dissemination of the axiomatic
method, the evolution of modern mathematical thinking.
János Bolyai’s work appeared as a short appendix
to Farkas Bolyai’s monumental two-volume book Tentamen
(Maros-Vásárhelyini, 1832–33), which summarized the
mathematical knowledge of the time. The Appendix
appeared in the first volume in 1832 but was also published
as a preprint in 1831. The Appendix was written in Latin, in
a concise, elegant and rigorous style. The text is divided
into forty-three sections and two major parts may be
distinguished in its contents. The first part is about
absolute geometry, the second about non-Euclidean
(hyperbolic) geometry. This copy of the Appendix belonged
to János Bolyai and contains his handwritten notes
and diagrams.