The middles ages stage began in the fourteenth century. It developed around studies in science in Europe. General notions were expressed in mechanical and geometrical forms, but like antiquity, each case of dependence between two quantities was defined verbally or graphically. Still no formulas were used.
During this time more functions were studied. All of the main trigonometrical functions were introduced and methods for tabulating them refined. The study of positive roots of cubic polynomials using conics advanced, and so did the analysis of accelerated motion.
They studied natural phenomena like heat, light, color, density, distance, and velocity. All of these have varying intensity. They also looked at functions of time. They determined the mean velocity of uniformly accelerated motion, an important concept for mechanics.
The ideas that developed centered around ideas of independent and dependent variable quantities, but definitions of these specifically were not given. A function was defined by a verbal description of its specific property or by a graph. All quantities and their relations were represented by geometrical forms, like straight lines and segments.