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.If you wish to remove this line, buy it now.Alfred North Whitehead - The Concept Of Nature.txtparallelograms in the four-dimensional manifold of event-particles.In parallelograms of the first type the two pairs of parallel sides are both ofthem pairs of rects.In parallelograms of the second type one pair of parallelsides is a pair of rects and the other pair is a pair of pointtracks.Inparallelograms of the third type the two pairs of parallel sides are both of thempairs of point-tracks.(128)The first axiom of congruence is that the opposite sides of any parallelogram arecongruent.This axiom enables us to compare the lengths of any two segmentseither respectively on parallel rects or on the same rect.Also it enables us tocompare the lengths of any two segments either respectively on parallelpoint-tracks or on the same point-track.It follows from this axiom that twoobjects at rest in any two points of a time-system are moving with equalvelocities in any other time-system along parallel lines.Thus we can speak ofthe velocity in a due to the time-system without specifying any particular pointin.The axiom also enables us to measure time in any time-system; but does notenable us to compare times in different time-systems.The second axiom of congruence concerns parallelograms on congruent bases andbetween the same parallels, which have also their other pairs of sides parallel.The axiom asserts that the rect joining the two event-particles of intersectionof the diagonals is parallel to the rect on which the bases lie.By the aid ofthis axiom it easily follows that the diagonals of a parallelogram bisect eachother.Congruence is extended in any space beyond parallel rects to all rects by twoaxioms depending on perpendicularity.The first of these axioms, which is thethird axiom of congruence, is that if ABC is a triangle of rects in any momentand D is the middle event-particle of the base BC, then the level through Dperpendicular to BC contains A when and only when AB is congruent to AC.Thisaxiom evidently expresses the symmetry of perpendicularity, and is the essence ofthe famous pons asinorum expressed as an axiom.The second axiom depending on perpendicularity,(129) and the fourth axiom of congruence, is that if r and A be a rect and anevent-particle in the same moment and AB and AC be a pair of rectangular rectsintersecting r in B and C, and AD and AE be another pair of rectangular rectsintersecting r in D and E, then either D or E lies in the segment BC and theother one of the two does not lie in this segment.Also as a particular case ofthis 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 twoaxioms the theory of congruence can be extended so as to compare lengths ofsegments on any two rects.Accordingly Euclidean metrical geometry in space iscompletely established and lengths in the spaces of different time-systems arecomparable as the result of definite properties of nature which indicate justthat particular method of comparison.The comparison of time-measurements in diverse time-systems requires two otheraxioms.The first of these axioms, forming the fifth axiom of congruence, will becalled the axiom of 'kinetic symmetry.' It expresses the symmetry of thequantitative relations between two time-systems when the times and lengths in thetwo systems are measured in congruent units.The axiom can be explained as follows: Let and be the names of twotime-systems.The directions of motion in the space of due to rest in a point ofis called the '-direction in ' and the direction of motion in the space of dueto rest in a point of a is called the '-direction in.' Consider a motion in thespace of a consisting of a certain velocity in the -direction of and a certainvelocity at right-angles to it.This motion represents rest in the space ofanother time-system -Page 48Easy PDF Creator is professional software to create PDF.If you wish to remove this line, buy it now.Alfred North Whitehead - The Concept Of Nature.txt(130) call it.Rest in will also be represented in the space of by a certainvelocity in the -direction in and a certain velocity at right-angles to this-direction.Thus a certain motion in the space of is correlated to a certainmotion in the space of , as both representing the same fact which can also berepresented by rest in.Now another time-system, which I will name , can befound which is such that rest in its space is represented by the same magnitudesof velocities along and perpendicular to the -direction in as those velocitiesin , along and perpendicular to the -direction, which represent rest in.Therequired axiom of kinetic symmetry is that rest in will be represented in bythe same velocities along and perpendicular to the -direction in as thosevelocities in along and perpendicular to the -direction which represent rest in.A particular case of this axiom is that relative velocities are equal andopposite.Namely rest in is represented in by a velocity along the -directionwhich is equal to the velocity along the -direction in which represents rest in.Finally the sixth axiom of congruence is that the relation of congruence istransitive.So far as this axiom applies to space, it is superfluous.For theproperty follows from our previous axioms.It is however necessary for time as asupplement to the axiom of kinetic symmetry.The meaning of the axiom is that ifthe time-unit of system is congruent to the time-unit of system , and thetime-unit of system is congruent to the time-unit of system , then the time-units of and are also congruent.By means of these axioms formulae for the trans-(131) -formation of measurements made in one time-system to measurements of thesame facts of nature made in another time-system can be deduced.These formulaewill be found to involve one arbitrary constant which I will call k.It is of the dimensions of the square of a velocity.Accordingly four casesarise [ Pobierz całość w formacie PDF ]