1 3 2 D O C . 3 1 I D E A S A N D M E T H O D S
is what we want to express when we say: the condition of intersection is covariant
under Euclidean transformations.
What has been said here in relation to conditions of intersection of two spherical
surfaces must hold quite generally for every mathematically formulated law that is
expressed on the basis of Euclidean geometry. All statements and laws must be co-
variant under Euclidean transformations in order to be meaningful from the point
of view of Euclidean geometry. 〈This is a general condition that all geometric and
physical relations must satisfy.〉
However, the Euclidean transformations, which, as we have said, have authori-
tative meaning for the construction of geometric and physical relations, are charac-
terized in the most simple manner by the following condition. Euclidean transfor-
mations are those that satisfy (identically) the equation
. (9)
This relation says geometrically that the square of the distance (measured with
the unit measuring rod) between two infinitesimally close points in space expresses
itself by coordinate differences and in an identical manner for both coordinate sys-
tems, specifically by means of the Pythagorean
theorem.3[29]
This distance is a
quantity independent of the coordinate system, an “invariant,” its square is the
“fundamental invariant of Euclidean transformations.”
Geometry and physics teach, especially in vector theory, how systems of equa-
tions are to be formed such that they are covariant under Euclidean transforma-
tions.
Again we return to the theory of special relativity. According to this theory—as
we have seen—the laws of nature are structured so that they are covariant under
Lorentz transformations (mathematical expression of the theory of special relativ-
ity). However, these transformations are characterized generally by satisfying iden-
tically the equation
. (6a)
It strikes the eye how similar this condition is to the one for the Euclidean trans-
formations (9). The similarity becomes more complete when we introduce in place
of the time coordinate the imaginary coordinate . Replacing by
, putting
3
This shows the fundamental importance of this theorem of our geometry.
dx1
2
dx2
2
dx3
2
+ +
dx1′2 dx2′2 dx3′2
+ + =
dx2 dy2 dz2 c2dt2 – + + dx′2 dy′2 dz′2 –c2dt′2 + + =
[p. 18]
t –1ct x y z t , , ,
x1, x2, x3, x4