Descartes took perhaps the greatest mathematical step in the realm of applied
mathematics in the development of the graphical representation of motion by the
use of so-called Cartesian co-ordinates. Descartes elucidated the goal toward
which his antecedents had climbed: the fundamental correspondence between number
and form. The trend of medieval mathematics had been to isolate the two,
assuming that form was unrelated to the mathematics of quantities and equations.
Descartes, by uniting the two realms of mathematics, paved the way for the
explanation of the motions of heavenly bodies, the effects of gravity on
projectiles, and many more phenomena that had previously been described but
never explained in the clear logic of mathematics. It is possible that the
application of algebraic methods to the geometry of form and motion is the most
important step taken in the progress of the exact sciences.
Few mathematical advances had effects as immediate as the study of
optics. As the importance of observation of the natural
world had grown, scientists had constantly sought the magnification of their
observed subjects. However, these scientists had been long plagued by
imperfections in the manufacture of glass lenses, which blurred images due to
high refraction and low resolution. It was not long before the principles of
geometry were applied to the field of optics, and glass grinders and their
scientist clients soon benefited from the revelations gleaned from this
application, which informed glass grinders of the specific measurements and
shapes lenses should have in order to maximize their power and resolution. The
culmination of these efforts was the introduction of the telescope and
microscope by Galileo in 1609, both of which revolutionized natural science.
