Tertium Organum

and time 
should come into 
contact or be able to come into contact. All the points of a sheet of paper spread out on 
a table are separated from one another. But, if we lift the sheet off the table, we can 
fold it so as to bring any points we like into contact. If on one corner we write 'St 
Petersburg' and on another 'Madras', this will not prevent us from folding these corners 
together. Or, if on one corner the year 1812 is written, and on another the year 1912, 
these corners can also be made to touch. If the year on one corner is written in red ink 
and the ink is not yet dry, the figures may get imprinted on another corner. Then, if the 
sheet is once more opened out and placed on the table, to a man who does not know 
that it can be lifted off the table and folded in many different ways, it will appear quite 
incomprehensible how a figure on one corner could become imprinted on another 
corner. The possibility of any contact between distant points of the sheet will be 
incomprehensible for him and will remain incomprehensible for him so long as he 
thinks of the sheet in two-dimensional space only. As soon as he imagines the sheet in 
three-dimensional space, this possibility will become real and obvious for him. 
* C. H. Hinton, A 
New Era of Thought,
London, George Allen & Unwin, 1910, p. 
44. 

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Examining the relation of the fourth dimension to the three dimensions 
known to us, we must admit that our geometry is obviously inadequate for the 
investigation of higher space. 
It was pointed out earlier that a four-dimensional body is incommensurable 
with a three-dimensional one, just as a 
year
is incommensurable with 
St 
Petersburg.
It is quite clear why this is so. A four-dimensional body consists of an 
infinitely great number of three-dimensional bodies; therefore, they can have 
no common measure. In comparison with a four-dimensional body, a three­
dimensional body is 
analogous to a point
as compared with a line. 
And, as a point is 
incommensurable
with a line, as a line is 
incommensurable with a surface, as a surface is incommensurable with a solid 
- so a three-dimensional body is incommensurable with a four-dimensional 
one. 
It is also clear why the geometry of three dimensions is not sufficient to 
define the 
position
of the domain of the fourth dimension in relation to three­
dimensional space. 
Just as in one-dimensional geometry, i.e. on a line, it is impossible to define 
the 
position of
the surface of which the given line is a side; 
just as on the surface - two-dimensional geometry - it is impossible to define 
the position of the solid of which the given surface is a side, so in three­
dimensional geometry, in three-dimensional space, it is impossible to define 
four-dimensional space. Putting it briefly, as planimetry is inadequate for the 
study of questions of stereometry, so stereometry is inadequate for the study
of four-dimensional space. 
As a deduction from everything that has been said, it may be repeated that 
each point of our space is a cross-section of a line of a higher space, or as 
Riemann put it: the material atom is the entry of the fourth dimension into 
three-dimensional space. 
In order to come nearer to this problem of higher dimensions and higher 
space it is first of all necessary to understand the essence of the domain of 
higher dimensions and its properties as compared with the domain of three 
dimensions. Only then will it be possible to investigate this domain more 
precisely and find out the laws which operate in it. 
What is it that we have to understand? 
It seems to me that, before anything else, it is necessary to understand that 
here it is not a question of 
two
spatially different domains — or of two 
domains, one of which (again spatially, 'geometrically') 

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constitutes a part of the other - but of two modes of perception of the same 
one
world of one space.
Further, it is necessary to understand that all the objects known to us exist 
not only in the categories in which we perceive them, but in an infinite 
number of others in which we do not know, or are unable to know, how to 
sense them. So first of all we must learn to 
think 
of things in other categories, 
then represent them to ourselves as far as we can in these other categories. 
Then and then only we may develop the capacity for perceiving things in 
higher space, and of sensing 'higher space' itself. 
Or, perhaps, the first thing required is a direct perception of everything in 
the surrounding world that is not included within the framework of three 
dimensions, that exists outside the category of time and space - everything, 
therefore, that we are accustomed to regard as non-existent.