Applied math. Part 1: Integral by Bocconi
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Extra info for Applied math. Part 1: Integral
If this construction is made on a plane surface, we 29 Albert Einstein have an uninterrupted disposition in which there are six discs touching every disc except those which lie on the outside. [ On the spherical surface the construction also seems to promise success at the outset, and the smaller the radius of the disc in proportion to that of the sphere, the more promising it seems. But as the construction progresses it becomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is not possible, as it should be possible by Euclidean geometry of the plane surface.
The disc-shadows are not rigid figures. ” But what if the two-foot rule were to behave on the plane E in the same way as the discshadows 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.
If we call the discshadows rigid figures, then spherical geometry holds good on the plane E 31 Albert Einstein 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. ” But what if the two-foot rule were to behave on the plane E in the same way as the discshadows 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.