_l_/_l′_ = 1/√(1 - _v²_/_c²_) _l′_ = _l_(1 - _v²_/_c²_).
This hypothesis sounds rather phantastical. For the contraction is not
to be thought of as a consequence of the resistance of ether, but purely
as a gift from the skies, as a sort of condition always accompanying a
state of motion.
I shall show in our figure that Lorentz’s hypothesis is fully equivalent
to the new conceptions about time and space. Thereby it may appear more
intelligible. Let us now, for the sake of simplicity, neglect (_y_, _z_)
and fix our attention on a two dimensional world, in which let upright
strips parallel to the _t_-axis represent a state of rest and another
parallel strip inclined to the _t_-axis represent a state of uniform
motion for a body, which has a constant spatial extension (see fig. 1).
If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis
and _x′_ as the _x_-axis, then the second body will appear to be at
rest, and the first body in uniform motion. We shall now assume that the
first body supposed to be at rest, has the length _l_, _i.e._, the cross
section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is
the unit measuring rod upon the _x_-axis—and the second body also, when
supposed to be at rest, has the same length _l_, this means that, the
cross section Q′Q′ of the second strip has a cross-section _l_^· OC′,
when measured parallel to the _x′_-axis. In these two bodies, we have
now images of two Lorentz-electrons, one of which is at rest and the
other moves uniformly. Now if we stick to our original coordinates, then
the extension of the second electron is given by the cross section QQ of
the strip belonging to it measured parallel to the _x_-axis. Now it is
clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.