"Up to now, one as has not succeeded to find a rigourous
proof of that truth [Euclid’s axiom on parallels]. Those which were given may
be named only explanations, but do not deserve to be considered, in the full
sense, mathematical proofs. " (Nikolai I Lobachevsky, 1823)
"[…] with Newton's and Descartes' time, the whole
Mathematics, becoming Analytic, walked so rapid steps forward that they left
far behind themselves this study without which they already could do and which
had ceased to draw on itself that attention which it deserved before." (Nikolai
I Lobachevsky, 1829)
"The theory of parallels explained by us supposes that the
lines and the angles are in a dependence which, as we shall see later, nobody
is in a position to prove whether it is found in nature or not." (Nikolai I
Lobachevsky, "Elements of Geometry", 1830)
"In the nature we properly know only the movement,
without which the sensory impressions are impossible. So, every other concepts,
for instance, the Geometrical ones, are artificially produced by our intellect,
taken from the properties of movement; and that is why the space in itself,
separately, for us does not exist. In consequence there cannot be any
contradiction in our intellect when we admit that certain forces in nature follow
one Geometry and others another particular one of their own." (Nikolai I
Lobachevsky, "New Elements of Geometry with a complete theory of Parallels", cca 1835-1838)
"In geometry I find certain imperfections which I hold to be
the reason why this science, apart from transition into analytics, can as yet
make no advance from that state in which it came to us from Euclid.
As belonging to these imperfections, I consider the
obscurity in the fundamental concepts of the geometrical magnitudes and in the
manner and method of representing the measuring of these magnitudes, and
finally the momentous gap in the theory of parallels, to fill which all efforts
of mathematicians have so far been in vain." (Nikolai I Lobachevsky," Geometric researches on the theory of parallels", 1840)
"The new (non-Euclidean) geometry, whose foundation has been laid here, even if it does not exist in nature, nevertheless may exist in our imagination; and, while remaining unused for actual measurements, it opens a new, extensive field for the mutual application of geometry and analysis." (Nikolai I Lobachevsky)
"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." (Nikolai I Lobachevsky)
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