Tertium Organum

independent direction
we mean, in this case, a line lying at right 
angles to another line. 
Our geometry (i.e. the science of measuring 
the earth,
or matter in space)
knows 
only three
such lines which lie simultaneously at right angles to one 
another and are not parallel in relation to each other. 
Why are there only three and not ten or fifteen? 
This we do not know. 
Moreover, one other fact is significant - either by virtue of some 
mysterious quality of the universe, or because of the limitations of our mental 
apparatus, we cannot 
visualize 
more than three perpendiculars. 
But we say that space is infinite. Therefore, since the first condition of 
infinity is infinity 
in all directions
and in all possible respects, we must 
assume that space has an infinite number of dimensions, that is, assume the 
possibility of an infinite number of lines perpendicular and not parallel to one 
another. And in addition we have to assume that 
for some reason
we know 
only three of these lines. 
This is the aspect in which the question of higher dimensions presents itself 
to our ordinary consciousness. 
All the same, since we are incapable of constructing more than three 
perpendiculars, we are forced to admit that, even if the 
three-dimensionality
of 
our space is merely conditional, the 
limitedness of 
our space as regards 
geometrical possibilities is an unquestionable fact. But of course, if these 
properties of space are created by certain attributes of our own, then it 
follows that the limitation is also in ourselves. 
* C. H. Hinton, 
A New Era of Thought,
London, George Alien & Unwin, 1910. 

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No matter what this limitation depends on, the fact is that it exists. 
A given point can be the vertex of only eight independent tetrahedrons. 
From a given point only three perpendicular and non-parallel lines can be 
traced. 
Starting from this, we determine the 
dimensionality
of space by the number 
of lines it is possible to trace in it which would lie at right angles to one 
another. 
On a line there cannot be a perpendicular, that is, 
another line.
It is one­
dimensional space. 
On a surface, two perpendiculars are possible. It is two-dimensional space. 
In 'space', there are three perpendiculars. It is three-dimensional space.
The idea 
of the fourth dimension
arose from the assumption that, in 
addition to the three dimensions known to our geometry, there exists a fourth, 
for some reason inaccessible and unknown to us, i.e. that in addition to the 
three perpendiculars known to us a mysterious fourth perpendicular is 
possible. In practice this assumption is based on the consideration that the 
world contains many things and phenomena about whose 
real existence
there 
can be no doubt, but which are utterly beyond being measured in length, 
breadth and height and lie, as it were, outside three-dimensional space. 
We may take as 
really existing
that which produces a certain action, has 
certain functions, represents the cause of something else. 
That which 
does not exist
cannot produce any action, has no function, 
cannot be a cause. 
But there are different kinds of existence. There is the 
physical 
existence, 
recognized by actions and functions of a certain kind; and there is the 
metaphysical existence, recognized by its actions and its functions. 
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