The King of Infinite Space Read Online Free

thought a definition of equality, and so a way of eliminating a troublesome concept altogether. It is not clear that this maneuver confers any great benefits. Among the things true of A is surely that A is equal to itself. The concept destined to be disappeared has just been reappeared. This might suggest that equality cannot easily be eliminated in favor of the truth because it cannot be eliminated at all.
    Just so.
    E UCLID ’ S FOURTH COMMON notion expresses a criterion of identity, a principle by which triangles, circles, or straight lines may be judged the same. The idea is implicit in every theorem that Euclid demonstrates. It is of the essence. If the geometer cannot tell when two shapes are the same, he cannot tell when they are different, and if he cannot tell whether shapes are the same or different, of what use is he?
    Suppose now that two triangles are separated in space. They become coincident when one of them is moved so that it covers the other in such a way that the two figures are perfectly aligned. Nothing is left over, extrudes, or sticks out.
    Coincidence or superposition offers the geometer a rough-and-ready measure of sameness in shape. What isnot entirely obvious in all this rough-and-readiness is just how figures separated in space—a triangle here, another one there—can be moved through space so that their coincidence may be tested.
    The point emerges early in the Elements ; it emerges in Euclid’s fourth proposition:
    If two triangles have the two sides equal to the two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.
    Two triangles are equal, Euclid has affirmed, if they are congruent, and they are congruent if two of their sides are equal, along with the angles the sides subtend.
    The proof is simple in its notoriety, for Euclid deviates at once into the swamp of concepts that he has not analyzed and cannot justify: “If,” he says, “the triangle ABC be applied to the triangle DEF, and if the point A be placed on the point D and the straight line AB on DE, then the point B will coincide with the point E, because AB is equal to DE.”
    Euclid is at the podium. He has just pointed to his dust board with the tip of an outstretched finger. Beaming with satisfaction, he is about to say . . .
    When he is interrupted.
    â€”Applied by whom, Sir?
    One question.
    â€”Placed how, Professor?
    Another.
    â€”Coincide when, Maître ?
    A third.
    B OTH B ERTRAND R USSELL and David Hilbert thought that Euclid would have been better served had he accepted proposition four as an axiom instead of claiming it as a theorem. It is a policy, as Russell remarked in another context, that has all the advantages of theft over honest toil. Designating Euclid’s fourth proposition an axiom does not do much to diminish the sense that in moving things around on the blackboard, the geometer has undertaken something at odds with the rigor of Euclidean geometry. In a little book titled Leçons de géométrie élémentaire (Lessons of elementary geometry), the French mathematician Jacques Hadamard proposed that coincidence be subordinated to some catalog of the ways in which shapes in Euclidean space might move. If the Euclidean idea of coincidence is a theorem, it depends on assumptions that Euclid did not make; if an axiom, it makes those assumptions without defending them; and if based on some antecedent assessment of motions allowed Euclidean figures, then it is both.
    The distinction between the concrete and the abstract models of Euclidean geometry offers a nice place in which to watch this uneasiness emerge and then separate itself into a destructive dilemma.
    Does the idea of coincidence apply to the concrete or the abstract