Monday, August 30, 2010

What is the point? -- consider Euclid

A two-line poem by Chilean poet, Pablo Neruda (1904-73), found in my bilingual edition of Extravagaria, reminded me of the poetic nature of several of the opening expressions of Euclid's geometry.  Both of these follow:
     Point     by Pablo Neruda, translated by Alastair Reid

     There is no space wider than that of grief,
     there is no universe like that which bleeds.

     (and in the original Spanish)   Punto

     No hay espacio más ancho que el dolor,
     no hay universo como aquel que sangra.

The following statments are from The Thirteen Books of Euclid’s Elements (Book I), translated by Thomas L. Heath (Cambridge University Press, 1908).  Book 1 of The Elements begins with numerous definitions followed by five postulates. Then, before Euclid starts to prove theorems, he gives a list of common notions. The first few definitions and the common notions are:

   1 Geometric Definitions
      1. A point is that which has no part.
      2. A line is length without breadth.
      3. The extremities of a line are points.
      4. A straight line is a line which lies evenly with the points on itself.
           . . .
   3 Common Notions
      1. Things which 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 which coincide with one another are equal to one another.
      5. The whole is greater than the part.

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