We must try to surmount this barrier in the mind, and the patient reader will see that it is by no means a particularly difficult task. For this purpose we will first give our attention once more to the geometry of two-dimensional spherical surfaces. In the adjoining figure let _K_ be the spherical surface, touched at _S_ by a plane, _E_, which, for facility of presentation, is shown in the drawing as a bounded surface. Let _L_ be a disc on the spherical surface. Now let us imagine that at the point _N_ of the spherical surface, diametrically opposite to _S_, there is a luminous point, throwing a shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the sphere has its shadow on the plane. If the disc on the sphere _K_ is moved, its shadow _L'_ on the plane _E_ also moves. When the disc _L_ is at _S_, it almost exactly coincides with its shadow. If it moves on the spherical surface away from _S_ upwards, the disc shadow _L'_ on the plane also moves away from _S_ on the plane outwards, growing bigger and bigger. As the disc _L_ approaches the luminous point _N_, the shadow moves off to infinity, and becomes infinitely great. Now we put the question, What are the laws of disposition of the disc-shadows _L'_ on the plane _E_? Evidently they are exactly the same as the laws of disposition of the discs _L_ on the spherical surface. For to each original figure on _K_ there is a corresponding shadow figure on _E_. If two discs on _K_ are touching, their shadows on _E_ also touch. The shadow-geometry on the plane agrees with the the disc-geometry on the sphere. If we call the disc-shadows rigid figures, then spherical geometry holds good on the plane _E_ with respect to these rigid figures. Moreover, the plane is finite with respect to the disc-shadows, since only a finite number of the shadows can find room on the plane. At this point somebody will say, "That is nonsense. The disc-shadows are _not_ rigid figures. We have only to move a two-foot rule about on the plane _E_ to convince ourselves that the shadows constantly increase in size as they move away from _S_ on the plane towards infinity." But what if the two-foot rule were to behave on the plane _E_ in the same way as the disc-shadows _L'_? It would then be impossible to show that the shadows increase in size as they move away from _S_; such an assertion would then no longer have any meaning whatever. In fact the only objective assertion that can be made about the disc-shadows is just this, that they are related in exactly the same way as are the rigid discs on the spherical surface in the sense of Euclidean geometry.