uld be made rigorous and comprehensive
independently of the specific content of axiom and definition was an
impossible one for the Greek, because such a suggestion could be made
only on the presupposition of a number theory and an algebra capable of
stating a continuum in terms which are independent of the sensuous
intuition of space and time and of the motion that takes place within
space and time. In the same fashion mechanics came back to fundamental
generalizations of experience with reference to motions which served as
axioms of mechanics, both celestial and terrestrial: the assumptions of
the natural motion of earthly substances to their own places in
straight lines, and of celestial bodies in circles and uniform
velocities, of an equilibrium where equal weights operate at equal
distances from the fulcrum.

The incommensurable of Pythagoras and the paradoxes of Zeno present the
"no thoroughfares" of ancient mathematical thought. Neither the
continuum of space nor of motion could be broken up into ultimate units,
when incommensurable ratios existed which could not be expressed, and
when motion refused to be divided into positions of space or time since
these are functions of motion. It was not until an algebraic theory of
number led mathematicians to the use of expressions for the irrational,
the minus, and the imaginary numbers through the logical development of
generalized expressions, that problems could be formulated in which
these irrational ratios and quantities were involved, though it is also
true that the effort to deal with problems of this character was in no
small degree responsible for the development of the algebra. Fixed
metaphysical assumptions in regard to number, space, time, motion, and
the nature of physical objects determined the limits within which
scientific investigation could take place. Thus though the hypothesis of
Copernicus and in all probability of Tycho Brahe were formulated by
Greek astronomers, their physical doctrine was unable to use them
because they were in flagrant contradiction with the definitions the
ancient world gave to earthly and celestial bodies and their natural
motions. The atomic doctrine with Democritus' thoroughgoing undertaking
to substitute a quantitative for a qualitative conception of matter
with the location of the qualitative aspects of the world in the
experience of the soul appealed only to the Epicurean who used the
theory as an exorcism to drive out of th