CHAPTER XII
NORMALITY AND CONGRUENCE
47. Normality. 47.1 A point-track will be said to be ‘normal’ to the moments of the time-system in the space of which it is a point.
A matrix is said to be ‘normal’ to the moments which are normal to any of the point-tracks which it contains.
Consider an event-particle P and a matrix m which contains P. Let α, β, γ, . . . be the collinear set of time-systems whose points lie in or are parallel to the matrix m. Let Pα, Pβ, Pγ, . . . be the moments of the time-systems α, β, γ, . . . which contain P. Then the levels Pαβ, Pβγ, . . . in which respectively Pα and Pβ, Pβ and Pγ, etc., intersect are identical, and the event-particle P is the sole event-particle forming the intersection of m and Pαβ. Also m intersects each of these moments Pα, and Pβ and Pγ, etc., in rects rα, rβ, rγ, etc., respectively. The level Pαβ and the matrix m are said to be mutually ‘normal.’ It will be noted that any two time-systems, α and β, determine one level and one matrix which are mutually normal and each contain a given event-particle. Corresponding to any level containing P there is one matrix normal to it at P; and corresponding to any matrix containing P there is one level normal to it at P.
If l and m be a level and a matrix normal to each other, then the rects in l will be called normal to the rects and point-tracks in m. A pair of rects which are normal to each other will also be called ‘perpendicular’ or ‘at right-angles.’ Two point-tracks can never be
