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It simply marks the fact that our congruence determination embraces both times and spaces in one universal system, and therefore if two arbitrary units are chosen, one for all spaces and one for all times, their ratio will be a velocity which is a fundamental property of nature expressing the fact that times and spaces are really comparable.
When the segment between two points A and B is congruent to that between the two points C and D, the quantitative measurements of the two segments are equal. The equality of the numerical measures and the congruence of the two segments are not always clearly discriminated, and are lumped together under the term equality. But the procedure of measurement presupposes congruence.
Measurement presupposes for its possibility nature as a simultaneity, and an observed object present then and present now. In other words, measurement of extended nature requires some inherent character in nature affording a rule of presentation of events. Furthermore congruence cannot be defined by the permanence of the measuring rod.
Suppose that with some expositors we cut out the reference to familiar velocities such as the rate of rotation of the earth. We are then driven to admit that there is no meaning in temporal congruence except that certain assumptions make the laws of motion true. Such a statement is historically false.
Also as a particular case of this axiom, if AB be perpendicular to r and in consequence AC be parallel to r, then D and E lie on opposite sides of B respectively. By the aid of these two axioms the theory of congruence can be extended so as to compare lengths of segments on any two rects.
The second axiom depending on perpendicularity, and the fourth axiom of congruence, is that if r and A be a rect and an event-particle in the same moment and AB and AC be a pair of rectangular rects intersecting r in B and C, and AD and AE be another pair of rectangular rects intersecting r in D and E, then either D or E lies in the segment BC and the other one of the two does not lie in this segment.
We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced. The theory of perpendicularity in the timeless space of any time-system α follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let ρ be any rect in the moment M of α and let λ be a level in M which is perpendicular to ρ.
Currently, Desert Storm and the liberation of Kuwait in 1991 have been taken as the examples that confirm the validity of the doctrine of overwhelming or decisive force and of ensuring that both strategic objectives and tactical methods were in congruence.
But in general it is inferred by a process of measurement depending on our direct sense-awareness of selected cases and a logical inference from the transitive character of congruence. Congruence depends on motion, and thereby is generated the connexion between spatial congruence and temporal congruence. Motion along a straight line has a symmetry round that line.
Perforce she had overborne reflection; one torment coming upon another had occupied her with mere endurance; it was as though a ruthless hand tore from her shred after shred of the fair garment in which she had joyed to clothe herself, while a voice mockingly bade her be in congruence with the sordid shows of the world around.
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