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The topic for this lecture is the continuation of the task of explaining the construction of spaces as abstracts from the facts of nature. It was noted at the close of the previous lecture that the question of congruence had not been considered, nor had the construction of a timeless space which should correlate the successive momentary spaces of a given time-system.
Uniformity in change is directly perceived, and it follows that mankind perceives in nature factors from which a theory of temporal congruence can be formed. The prevalent theory entirely fails to produce such factors. The mention of the laws of motion raises another point where the prevalent theory has nothing to say and the new theory gives a complete explanation.
It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt.
The axiom asserts that the rect joining the two event-particles of intersection of the diagonals is parallel to the rect on which the bases lie. By the aid of this axiom it easily follows that the diagonals of a parallelogram bisect each other. Congruence is extended in any space beyond parallel rects to all rects by two axioms depending on perpendicularity.
In the second place, the deliberate judgment of any rationally minded individual is entitled to respect as a source of truth. Conflict must in the last analysis be overcome by the congruence of impartial minds.
The first axiom of congruence is that the opposite sides of any parallelogram are congruent. This axiom enables us to compare the lengths of any two segments either respectively on parallel rects or on the same rect. Also it enables us to compare the lengths of any two segments either respectively on parallel point-tracks or on the same point-track.
Nevertheless, while I must acknowledge this insuperable difference between the Progress we can make our end or ideal and the Progress we believe that in ourselves and around us we apprehend, I still would lay renewed stress upon the congruence and affinity of the two, and urge that the perception of the one the Progress without us and the pursuit of the other the Progress within us support and fertilize each the other.
I cannot see the answer to either of these contentions provided that you admit the materialistic theory of nature. With this theory nature at an instant in space is an independent fact. Thus we have to look for our preeminent congruence relation amid nature in instantaneous space; and Poincaré is undoubtedly right in saying that nature on this hypothesis gives us no help in finding it.
We then enquire about congruence and lay down the set of conditions or axioms as they are called which this relation satisfies.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes in fact, a complete theory of non-metrical geometry.
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