consideration of
figures? For the ancients, geometry was a purely static science. Figures
were given to it at once, completely finished, like the Platonic Ideas.
But the essence of the Cartesian geometry (although Descartes did not
give it this form) was to regard every plane curve as described by the
movement of a point on a movable straight line which is displaced,
parallel to itself, along the axis of the abscissae--the displacement of
the movable straight line being supposed to be uniform and the abscissa
thus becoming representative of the time. The curve is then defined if
we can state the relation connecting the space traversed on the movable
straight line to the time employed in traversing it, that is, if we are
able to indicate the position of the movable point, on the straight line
which it traverses, at any moment whatever of its course. This relation
is just what we call the equation of the curve. To substitute an
equation for a figure consists, therefore, in seeing the actual position
of the moving points in the tracing of the curve at any moment whatever,
instead of regarding this tracing all at once, gathered up in the unique
moment when the curve has reached its finished state.

Such, then, was the directing idea of the reform by which both the
science of nature and mathematics, which serves as its instrument, were
renewed. Modern science is the daughter of astronomy; it has come down
from heaven to earth along the inclined plane of Galileo, for it is
through Galileo that Newton and his successors are connected with
Kepler. Now, how did the astronomical problem present itself to Kepler?
The question was, knowing the respective positions of the planets at a
given moment, how to calculate their positions at any other moment. So
the same question presented itself, henceforth, for every material
system. Each material point became a rudimentary planet, and the main
question, the ideal problem whose solution would yield the key to all
the others was, the positions of these elements at a particular moment
being given, how to determine their relative positions at any moment. No
doubt the problem cannot be put in these precise terms except in very
simple cases, for a schematized reality; for we never know the
respective positions of the real elements of matter, supposing there are
real elements; and, even if we knew them at a given moment, the
calculation of their positions at another moment would generally requi