and showed that the arbitrary function formula_4 must be unity for all formula_5 (Lorentz had set formula_6 by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination formula_7 is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing formula_8 as a fourth imaginary coordinate, and he used an early form of four-vectors. Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was Hermann Minkowski who worked out the consequences of this notion in 1907.