184 
HINTON. 
We can represent the plane being and his object by figures 
cut out of paper, which slip on a smooth surface. The 
thickness of these bodies must be taken as so minute that 
y their extension in the third dimension 
escapes the observation of the plane 
being, and he thinks about them as if 
they were mathematical plane figures 
in a plane instead of being material 
x bodies capable of moving on a plane 
surface. Let A x, A y be two axes and 
ABCD a square. As far as move- 
ments in the plane are concerned, the square can rotate 
about a point, A, for example. It cannot rotate about a side 
such as A C. 
But if the plane being is aware of the existence of a third 
dimension he can study the movements possible in the ample 
space, taking his figure portion by portion. 
His plane can only hold two axes. But, since it can hold 
two, he is able to represent a turning into the third dimen¬ 
sion if he neglect one of his axes and represent the third 
axis as lying in his plane. He can make a drawing in his 
plane of what stands up perpendicular^ from his plane. 
Let A z be the axis, which stands per¬ 
pendicular to his plane at A. He can 
draw in his plane two lines to represent 
the two axes, A x and A z. Let Fig. 2 
be this drawing. Here the 2 axis has 
taken the place of the y axis, and the —- 
plane of A x A 2 : is represented in his 
plane. In this figure all that exists of 
Br 
B 
Fig. 
the square ABCD will be the line A B. 
The square extends from this line in the y direction, but 
more of that direction is represented in Fig. 2. The plane 
being can study the turning of the line A B in this diagram. 
It is simply a case of plane turning around the point A. 
The line A B occupies intermediate portions like A B l and 
after half a revolution will lie on A x produced through A. 
