space and time.  This conception must have been dealt
　　   a severe blow by the Pythagorean  discovery that  the
　　   hypotenuse　　 of   a   unit   right   triangle　　 was
　　   incommensurable  with  its sides, and that  therefore
　　   there  could be no one unit, however  small, of which
　　   both  could  be composed.  Attempts  to resolve  this
　　   difficulty led to great ambiguity as to the nature of
　　   atoms,  which  varied  according   to  context   from
　　   entities  of  definite  magnitude   to  dimensionless
　　   points and instants. The Pythagoreans maintained both
　　   that  the world  was composed  of atoms  and that any
　　   magnitude was infinitely divisible, No one definition
　　   of the atom would  suffice.  If it were taken to have
　　   definite  magnitude, then there would  be lines which
　　   could  not  be bisected, and  no magnitude  would  be
　　   infinitely divisible; if, on the other hand, the atom
　　   were　　 made   dimensionless　　 to   give　　 infinite
　　   divisibility, no quantity  of such atoms  could  ever
　　   add  up  to  any  magnitude  at  all.   According  to
　　   Brumbaugh, Zeno's  Paradoxes  were designed  to bring
　　   out the inherent absurdities of such a world view and
　　   to show that, however one interpreted  this position,
　　   whichever  of its premises one adopted, no account of
　　   motion   could   be   given  which  did  not  end  in
　　   absurdity, Whether  space and time were atomistic  or
　　   infinitely  divisible,  no  intelligible  account  of
　　   motion through them was possible.
　　　　  There are four possible combinations  here: Space
　　   might be continuous  (that is, infinitely  divisible)
　　   and  time  discrete  (that  is, composed  of extended
　　   minims or atoms); or space might be discrete and time
　　   continuous;  or both might be continuous;  or, again,
　　   both  might  be  discrete.   The  Bisection  Paradox,
　　   Achilles and the Tortoise, the Arrow, and the Stadium
　　   are designed  to refute, respectively, each  of these
　　   possibilities.  Each Paradox  depends  for its effect
　　   upon its proper  suppressed  premise  concerning  the
　　   nature of space and time.
　　　　  The  Bisection  Paradox  assumes  that  space  is
　　   continuous  (infinitely  divisible) and time discrete
　　   (atomistic). Zeno presents it as follows:

　　　　　　　　　　　　  p.283

　　  

　　   ...  The first asserts the non-existence of motion on