100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Development of the Concepts of Space, Time and Space-Time 19And although a time and a line are [mutually] incomparable inquantity, still there is no ratio found as existing between time andtime, which is not found among lines, and vice versa. (A Treatiseon the Configuration of Qualities and Motions).It was not until 1698 that Pierre Varignon combined spatial and temporalintervals in a single diagram. He commented:Space and time being heterogeneous magnitudes, it is not reallythem that are compared together in the relation called velocity,but only the homogeneous magnitudes that express them, which are. . . either two lines or two numbers, or any two other homogeneousmagnitudes that one wishes (Mémoire of 6 July 1707).Forty years later, Jean le Rond D’Alembert was even more explicit:One cannot compare together two things of a different nature, suchas space and time; but one can compare the relation of two portionsof time with that of the parts of space traversed. . . . One mayimagine a curve, the abcissae of which represent the portions oftime elapsed since the beginning of the motion, the correspondingordinates designating the corresponding spaces traversed duringthese temporal portions: the equation of this curve expresses, notthe relation between times and spaces, but if one may so put it,the relation of relation that the portions of time have to their unitto that which the portions of space traversed have to theirs (Traitédu dynamique, 1743).The only twentieth-century mathematician I have found who emphasizesthe dimensional nature of physical quantities is J. A. Schouten, the eminentdifferential geometer. In Ref. 8, he distinguished between geometrical andphysical quantities, pointing out (p. 126) that:quantities in physics have a property that geometrical quantitiesdo not. Their components change not only with transformations ofcoordinates but also with transformations of certain units.Constantly bearing in mind this difference between mathematical and physicalquantities can help avoid the fetishism of mathematics.