The King of Infinite Space Read Free Page A

raison, c’est de connaître qu’il y a une infinité de choses qui la surpassent. (The last step of reason is to grasp that there are infinitely many things beyond reason).
    â€”P ASCAL
    â€œE UCLID ALONE ,” E DNA St. Vincent Millay once wrote, “has looked on beauty bare.” It is the first line in a sonnet of the same name. Literary critics are often embarrassed by the sonnet, and mathematicians by Edna St. Vincent Millay. Euclid alone ? Still, the idea that “Euclid alone looked on beauty bare” elegantly draws attention to the nakedness of inference exhibited by every Euclidean proof. It is something rarely seen beyond mathematics—this hidden, if somewhat lurid, power of a Euclidean proof to compel fascination. Up go the axioms on the blackboard; down come the theorems. Students and readers alike are encouraged to think of the display as something stirring. And it is. So much of ordinary argument and inference is fully clothed.
    But this way of presenting Euclid and the Elements imposes a gross distortion on Euclid’s thoughts: it allows the staged drama of his proofs to stand for the grandeur of his system as a whole. Euclid meant his proofs to be grasped against the background of his common notions and definitions. In almost every proof, he appeals to his own common notions and, in many proofs, either to his definitions or to ideas that follow naturally from his definitions. Beyond any of this, there are Euclid’s ideas about space and human agency and the exaltation of geometry that is so conspicuous a feature of his thoughts. Focus, control, and tension—they are there in Euclid’s proofs, but these moments, as any athlete knows, do not appear as isolated, brief, bursting miracles. They are not isolated at all, and they are not miracles either. They are grounded in Euclid’s meditations about what may be supposed and what not, and how difficult ideas may be defined or, at least, exposed. In all this, the master, unbending to explain himself, remains entirely in character, his severity undiminished, no word wasted, as prudent, compact, and tight as the stretched skins on which he wrote.
    E UCLID ’ S COMMON NOTIONS represent the “beliefs on which all men base their proofs.” The words are Aristotle’s, but theidea that there are beliefs on which all men base their proofs must itself have been one of them, for Euclid appropriated the idea without hesitation and without argument.
    There are five common notions in all:
    1.Things that are equal to the same thing are also equal to one another.
    2.If equals be added to equals, the wholes are equal.
    3.If equals be subtracted from equals, the remainders are equal.
    4.Things that coincide with one another are equal to one another.
    5.The whole is greater than the part.
    These principles convey an air of what is obvious. They have authority. No one either in Euclid’s time or our own is proposing that if equals are added to equals, the result might be un equal. A surprising delicacy is nonetheless required to say just what these principles mean. It is a delicacy that Euclid did not possess. This might suggest that Euclid’s conviction that these beliefs are common represented on his part a willingness to repose his confidence in things he could neither explain nor justify. To say as much involves no rebuke. If Euclid could neither explain nor justify the common beliefs that he invoked, we can do as little, or as much, with respect to our own. It wasEuclid’s genius to grasp that whatever the powers of his geometrical system, it did rest on certain common beliefs.
    It was Euclid’s business to say what those beliefs were.
    And our business to say what they mean.
    E QUALITY IS AN indispensable idea. It is like water to the fish—everywhere at once, but easy to ignore and difficult to define. To say of two things that they are equal is always false, and to say of one thing that it is