A main condition necessary for the advancement of physics and astronomy that progressed during the Scientific Revolution was the advance of mathematics, which allowed the proof of abstract theories and provided a more logical method for attacking the Aristotelian system. During the late sixteenth century, a French lawyer, Francois Viete, was among the first to use letters to represent unknown quantities. In 1591 and after, he applied this algebraic method to geometry, laying the foundation for the invention of trigonometry. The Fleming Simon Stevin also worked with geometry during the late sixteenth century, applying it to the physics of incline planes and the hydrostatic surface tension of water. Additionally, he introduced the decimal system of representing fractions, an advance which greatly eased the task of calculation.
However, perhaps the most important mathematical advance of the early period of the Scientific Revolution was the invention of logarithms in 1594 by John Napier of Scotland. Napier spent the next 20 years of his life developing his theory and computing an extensive table of logarithms to aid in calculation. In 1614, he published Description of the Marvelous Canon of Logarithms, which contained the fruits of these labors.
Johannes Kepler also did a great deal of work in geometry, which proved significant to his subsequent work in astronomy. In 1637, Rene Descartes published Geometry, in which he describes how geometry relates to motion and showed that at any moment the position of a point can be defined by its relation to surrounding planes or reference. The most well known application of this theory is the use of a curve on a graph to represent the motion of an object, which could then be defined by a mathematical equation that would grant insight into the forces at work on the object.
Further progress in mathematics was made by Oxford professor John Wallis. His first work, Arithmetica Infinitum, published in 1655, set the stage for the invention and development of differential calculus. Wallis' became one of Isaac Newton's major influences. Wallis was the first mathematician to apply mathematics to the operation of the tides, and also invented the symbol used to denote infinity.
Many mathematicians applied their knowledge to the study of optics, a field that had garnered great interest since the Middle Ages. The advances made in this field, including the development of techniques for higher resolution, led to the better construction of optical instruments, such as the telescope, which played a large part in the later work of Galileo.
Mathematics developed as a response to the demands of the sciences, which grew up in the late sixteenth century. The thinkers of the early Scientific Revolution had provided their descendents with a broad framework of new philosophies, hypotheses, and qualitative observations, all of which pointed to a revolution in thought. However, the old order was at first easily preserved in the face of this onslaught, in part due to the lack of substance to back up the theories of such thinkers as Nicolas Copernicus and Giordano Bruno. Though these scientists sensed that their hypotheses were correct, and strongly believe in them on their own, it was difficult to bring their theories to the position of respect they deserved without the benefit of clear and logical evidence. The realm of mathematics potentially offered this clear and logical evidence. The scientists of the early Scientific Revolution knew that there were forces acting on the physical world that would explain the phenomena they observed, but they had no way to quantify these forces and apply them to the geometry of the physical world. Mathematicians strove to solve this problem with the development of trigonometry and the application of new mathematical theorems to the physics of motion. Armed with these tools, the scientists of the early Scientific Revolution began to back up their hypotheses with mathematical proofs were nearly beyond question.
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.