Tertium Organum

It is more than possible that four-dimensional space is the distance between 
a number of solids, separating yet at the same time binding into some 
incomprehensible whole, those solids which to us appear to be separate from 
one another. 
Moreover, we regard a point as a 
section
of a line; a line as a section of a 
surface; a surface as a section of a solid. 
By analogy with this it may be possible to regard a solid (a cube, a sphere, 
a pyramid) as a 
section
of a four-dimensional body; and the whole of three­
dimensional space as a section of four-dimensional space. 
If every three-dimensional body is the section of a four-dimensional one, 
then every point of a three-dimensional body is the section of a four­
dimensional line. An 'atom' of a physical body may be regarded, 
not as 
something material,
but as the intersection of a four-dimensional line by the 
plane of our consciousness. 
The view of a three-dimensional body as a section of a four-dimensional 
one leads us to the thought that many three-dimensional bodies, which appear 
separate for us, may be 
sections or parts of 
one four-dimensional body.
A simple example will illustrate this idea. If we imagine a horizontal plane, 
intersecting the top of a tree in a direction parallel to the earth, then 
on this 
plane 
the sections of the branches will appear separate and quite unconnected 
with one another. And yet in our space, from our point of view, these are 
sections of the branches 
of one 
tree, together forming one top, fed by one 
common root and casting one shadow. 
Or again, another interesting example illustrating the same idea is 

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given by the theosophical writer, C. W. Leadbeater, in one of his books. If we 
touch the surface of a table with our five fingertips of one hand, there will be 
then on the surface of the table only five circles, and on 
this surface
it is 
impossible to have any idea either of the hand or of the man to whom the 
hand belongs. There will be five 
separate 
circles on the table's surface. How, 
from these, is it possible to picture a man, with all the richness of his physical 
and psychological life? It is impossible. Our relation to the four-dimensional 
world may be exactly the same as the relationship between that consciousness 
which sees the five circles on the table and 
the man.
We see only 'fingertips'; 
that is why the fourth dimension is incomprehensible for us. 
In addition, we know that it is possible to draw an image of a three­
dimensional body on a plane, that it is possible to draw a cube, a polyhedron, 
or a sphere. But it will not be a real cube or a real sphere, but only the 
projection of a cube or a sphere on a plane. So it may be that we are justified 
in thinking that the three-dimensional bodies we see in our space are 
images, 
so to
speak, of four-dimensional bodies, incomprehensible for us. 

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CHAPTER 4 
In what direction may the fourth dimension lie? What is motion? Two kinds of 
movement - movement in space and movement in time -contained in every motion. 
What is time? Present past and future. Wundt on sense-cognition. Groping through 
life. Why we do not see the past and the future. A new extension in space and motion 
in that space. Two ideas contained in the concept of time. Time as the fourth 
dimension of space. Impossibility of understanding the idea of the fourth dimension 
without the idea of motion. The idea of motion and 'time-sense'. 'Time-sense' as the 
limit (surface) of space sense. Riemann's idea of the translation of time into space in 
the fourth dimension. Hinton on the law of surfaces. 'Ether' as a surface. 
From the analogy between the relation of lower dimensional figures to higher
dimensional figures we have established the fact that a four-dimensional body 
may be regarded as the trace of the movement of a three-dimensional body in 
a direction not contained in it, i.e. that the direction of motion in the fourth 
dimension lies outside all the directions possible in a three-dimensional 
space. 
What can this direction be? 
In order to answer this question we must see whether we know of any 
movement in a direction not contained in three-dimensional space. 
We know that every movement in space is accompanied by what we may
call