L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 4 7 Immediately following, he wrote the equation of propagation of waves, and by substi- tuting the coordinates and the time of one system with those of the other, through Lorentz’s relationships, he demonstrated the truth of that assertion, thus ending the lecture. On concluding his lecture, Professor Einstein received prolonged applause, and imme- diately thereafter he left the lecture hall, accompanied by several professors, with whom he conversed extensively about various matters related to his theories. Professor Einstein Gave His Third Lecture on Relativity Yesterday [1 April 1925] Minkowski’s Universe: A Fundamental Invariant Dr. Einstein began by recalling that in his previous lecture the Lorentz transformation equations had been presented and their content revealed: the relativity of space and time. Furthermore, the forms of the natural laws, he added, are identical for all systems that move relatively with rectilinear and uniform movement. They do not depend, therefore, on the ob- server, or, stated differently, they are invariant. We must now consider such transformations in a more general way, he said. Minkowski, the sage continued, has made it evident that they contain an invariant, fundamental relation- ship between distances and durations that are accessible to observers belonging to the dif- ferent systems. “The square of the distance between the locations where two events take place, minus the square of the speed of light times the square of the time elapsed: between them they yield exactly the same value.” Such a result should not be surprising, he went on, since, as the expressions of the coor- dinates and the time with which the laws of natural phenomena are represented in the dif- ferent systems are identical, despite the fact that the distances and durations achieved in them are different, it is clear that it was feasible to expect the existence of an invariant relationship between these. With this as a given, he said, let us proceed to Minkowski’s demonstration. And he add- ed: For each event there is a place and an instant of time that are perfectly defined with re- spect to a particular observer, and since the location of a point in space, with reference to a system in which that point is at rest, is represented by three numbers (the coordinates), it is clear that the localization and history of the events require the knowledge of four parame- ters: the three spatial coordinates and time. If it were possible to represent geometrically the set of the dimensions that locate all the events in time, we would have a static image of the evolution of the world of phenomena (“The demonstration that an omnipresent intelligence would have to use in order to define the world of phenomena”). Minkowski, he continued, has formally developed the geometry of the continuum of such dimensions, a continuum that he calls “universe.” This universe is not a space, in the