, now that we have abandoned
the ether, no such support. We can no longer establish a system of
co-ordinates, like the one just mentioned, in a universal intermediate
matter, and if we were to arrive in one way or another at a definite
system of lines crossing each other in three directions, then we should
be able to use just as well another similar system that in respect to
the first moves this or that way. We should also be able to remodel the
system of co-ordinates in all kinds of ways, for example by extension
or compression. That in all these cases for fixed bodies that do not
participate in the movement or the remodelling of the system other
co-ordinates will be read off again and again is clear.

NEW SYSTEM OR CO-ORDINATES

What way Einstein had to follow is now apparent. He must--this
hardly needs to be said--in calculating definite, particular cases
make use of a chosen system of co-ordinates, but as he had no means
of limiting his choice beforehand and in general, he had to reserve
full liberty of action in this respect. Therefore he made it his aim
so to arrange the theory that, no matter how the choice was made, the
phenomena of gravitation, so far as its effects and its stimulation
by the attracting bodies are concerned, may always be described in
the same way--i.e., through comparisons of the same general form,
as we again and again give certain values to the numbers that mark
the sphere of gravitation. (For the sake of simplification I here
disregard the fact that Einstein desires that also the way in which
time is measured and represented by figures shall have no influence
upon the central value of the comparisons.)

Whether this aim could be attained was a question of mathematical
inquiry. It really was attained, remarkably enough, and, we may say, to
the surprise of Einstein himself, although at the cost of considerable
simplicity in the mathematical form; it appeared necessary for the
fixation of the field of gravitation in one or the other point in
space to introduce no fewer than ten quantities in the place of the
one that occurred in the example mentioned above.

In this connection it is of importance to note that when we exclude
certain possibilities that would give rise to still greater intricacy,
the form of comparison used by Einstein to present the theory is
the only possible one; the principle of the freedom of choice in
co-ordinates was the only one by which he needed to allow himsel