Pagina 168 di 212

In the first place, we imagine an existence in two dimensional

space. Flat beings with flat implements, and in particular flat

rigid measuring-rods, are free to move in a plane. For them

nothing exists outside of this plane: that which they observe to

happen to themselves and to their flat "things" is the

all-inclusive reality of their plane. In particular, the

constructions of plane Euclidean geometry can be carried out by

means of the rods e.g. the lattice construction, considered in

Section XXIV. In contrast to ours, the universe of these beings

is two-dimensional; but, like ours, it extends to infinity. In

their universe there is room for an infinite number of identical

squares made up of rods, i.e. its volume (surface) is infinite.

If these beings say their universe is "plane," there is sense in

the statement, because they mean that they can perform the

constructions of plane Euclidean geometry with their rods.