Tertium Organum

meta-geometry, 
and he connects the names of Lobachevsky, Gauss and other investigators of 
non-Euclidean geometry with metageometry.
Let us now examine how the theories of these scientists stand in relation to 
the questions we have raised. 
Hinton deduces his ideas from Kant and Lobachevsky. 

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Others, on the contrary, set Kant's ideas in opposition to those of 
Lobachevsky. Thus, Roberto Bonola, in 
Non-Euclidean Geometry, 
says that 
Lobachevsky's view of space is opposed to that of Kant. He says: 
The Kantian doctrine considered space as a subjective intuition, a necessary 
presupposition of every experience. Lobachevsky's doctrine was rather allied to 
sensualism and the current empiricism, and compelled geometry to take its place 
again among the experimental sciences!* 
Which view is correct and in what relation do Lobachevsky's ideas stand to 
our problem? The most correct answer would be: in no relation. Non-
Euclidean geometry is not 
metageometry,
and non-Euclidean geometry stands 
to metageometry in the same relation as does Euclidean geometry.
The results of all non-Euclidean geometry, which revalued the fundamental 
axioms of Euclid and found its fullest expression in the works of Bolyai, 
Gauss and Lobachevsky, are expressed in the formula: 
The axioms of a given geometry express the properties of a given 
space. 
Thus, plane geometry accepts all three Euclidean axioms, i.e.: 
1 A straight line is the shortest distance between two points. 
2 Any figure may be transferred to another place without interfering with 
its properties.
3 Parallel lines do not meet. (This last axiom is usually formulated 
differently according to Euclid.) 
In the geometry of a sphere or a concave surface only the first two axioms 
are true, for the meridians, parallel at the equator, meet at the poles.
In the geometry of an irregularly curved surface only the first axiom is true; 
the second (about the transfer of figures) no longer holds good, for a figure
taken from one place of an irregular surface may change when transferred to 
another place. And the sum of the angles of a triangle may be more or less 
than two right angles. 
Thus, 
axioms
express the difference in the properties of different kinds of 
surfaces. A geometric axiom is a law of a 
given
surface. 
* Roberto Bonola, 
Non-euclidean Geometry, a Critical and Historical Study of its 
Development,
Open Court Publishing Co., Chicago, 1912, pp. 92 and 93. 

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But what is a surface? 
Lobachevsky's merit lies in the fact that he found it necessary to revise the 
fundamental concepts of geometry. But he never went so far as to revalue 
them from Kant's point of view. Yet at the same time, he never argued 
against Kant in any sense. For Lobachevsky, as a geometrician, a 
surface
was 
merely a means for the generalization of certain properties upon which one or 
another geometric system was built, or the means for generalizing the 
properties of certain given lines. He probably never thought at all about the 
reality or the unreality of a surface. 
Thus, on the one hand, Bonola is quite wrong in ascribing to Lobachevsky
views opposed to those of Kant, and approaching 'sensualism' and 'the current 
empiricism'; while on the other hand, there are grounds for thinking that 
Hinton is quite subjective in ascribing to Lobachevsky and Gauss the 
inauguration of a new era in 
philosophy. 
Non-Euclidean geometry, including Lobachevsky's geometry, bears no 
relation to