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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.

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