The celestial sphere

There are two approaches to teaching astronomy: the first starts with the nearest objects and the best observable phenomena, the other starts with the universe and treats all the rest as its fragments. The first approach dominates. Usually a course in astronomy begins with the introduction of an object which has a very characteristic feature: it does not exist. This is the celestial sphere. To be more precise, over two thousand years many people, even very competent, insisted on its existence. After all, everyone could see it above on a clear night. This was the so-called sphere of constant stars. Nevertheless, over the last few hundred years the sphere was being mentioned with ever less conviction. Today the celestial sphere, even if nonexistent, remains the scenery of all heavenly phenomena - which is quite a banality - but it is also an object used to determine coordinates, to compute trigonometry, to represent all the questions concerning sunrises and sunsets, seasons of the year, precession, navigation etc. In short, it is the foundation of all spherical astronomy.

The celestial sphere is certainly not the only physically nonexisting object which turned out to be very useful and thus persists as an idea. There is an infinity of such objects in mathematics and in physics. The difference is that a few thousand years ago it was endowed with real existence (even if it was thought to be made of a material "not from this earth") and has been deprived of it quite recently. Has anyone ever claimed the material character of a straight line, a Lagrangian, a horned sphere or a wave function?