¹ Well before the invention of Calculus, Descartes had already refuted the Achilles, demonstrating that the series 10^-1 + 10^-2 + … + 10^-n + … is convergent and that sum is 1/9 (Letter to Clerselier, June or July 1646, A.T. IV, p. 442).

² In Riemannian space, the luminous rays which in ordinary geometric space are straight lines can be curves which intersect in more than one point. In the simple case of the hypersphere, all the "photons" leaving at a given instant from any point whatsoever will meet at the antipodes if the space is empty. Thus, on the surface of the earth, two travelers leaving the North Pole at the same time and moving along two different meridians at the same speed will meet at the South Pole, supposing that no psychological or material obstacle has intervened.

³ "Isotropy" and "anisotropy" are words borrowed from optics by geometry. A transparent crystal is "anisotropic" when its index of refraction varies according to the direction of the beam of light which passes through it. By analogy, a space is called anisotropic if the directions are not freely interchangeable as they are in the space of ordinary geometry. In speaking of the anisotropy of Time, Grünbaum means that the direction past → future is not interchangeable with the direction future → past.

⁴ Hermann Minkowski demonstrated in 1908 that the kinetic laws of a partial theory of relativity, so surprising when one envisages them within the framework of traditional, "Euclidian" and "Galileian" conceptions of Space and Time, are in fact the theorems of a geometry which relates Space and Time in a "Space-Time;" this "continuum" or this "diversity" in four dimensions possesses analogous me trical properties, but not identical to those of the space of ordinary geometry.

⁵ "Thus the idea inherent to causality, that that which is anterior is the deter mining reason for what follows, and not vice versa impresses on our judgement of probability a direction separate from that of Time." H. Weyl, Philosophy of Math ematics and Natural Science, Princeton, N.J., 1949, p. 203.

⁶ "Time is not a real process, an effective succession that I limit myself to record ing. It is born from my relationship with things." M. Merleau-Ponty, Phénoméno logie de la perception, Paris, 1945, p. 471.