The fourth dimension

58 THE FOURTH DIMENSION

from the familiar intuitions, was so difficult that almost any other hypothesis was mere easy of acceptance, and when Beltrami showed that the geometry of Lobatchewsky and Bolyai was the geometry of shortest lines drawn on certain curved surfaces, the ordinary definitions of measurement being retained, attention was drawn to the theory of a higher space. An illustration of Beltrami’s theory is furnished by the simple consideration of hypothetical

beings living on a spherical surface. Let ascp be the equator of a globe, and ap, BP; meridian lines drawn to the pole, P.

P The lines AB, AP, BP would seem to be Ki perfectly straight to a person moving D c on the surface of the sphere, and

\J unconscious of its curvature. Now

Ap and BP both make right angles

Pp with AaB. Hence they satisfy the

Pig. 33. definition of parallels. Yet they

meet in Pp. Hence a being living on a spherical surface,

and unconscious of its curvature, would find that parallel

lines would meet. He would also find that the angles

in a triangle were greater than two right angles. In

the triangle pas, for instance, the angles at a and B

are right angles, so the three angles of the triangle PAB are greater than two right angles.

Now in one of the systems of metageometry (for after Lobatchewsky had shown the way it was found that other systems were possible besides his) the angles of a triangle are greater than two right angles.

Thus a being on a sphere would form conclusions about his space which are the same as he would form if he lived on a plane, the matter in which had such properties as

_ are presupposed by one of these systems of geometry. J Beltrami also discovered a certain surface on which there / could be drawn more than one “straight” line through a